JEE Mains JEE Advance BITSAT
JEE Mains | PYQ

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The equations of the sides \( AB \), \( BC \) and \( CA \) of a triangle \( ABC \) are: \( 2x + y = 0 \), \( x + py = 21a \) \( (a \neq 0) \) and \( x - y = 3 \) respectively. Let \( P(2, a) \) be the centroid of \( \triangle ABC \). The \( (BC)^2 \) is equal to:

A triangle is formed by \( x \)-axis, \( y \)-axis and the line \( 3x+4y=60 \). Then the number of points \( P(a,b) \) which lie strictly inside the triangle, where \( a \) is an integer and \( b \) is a multiple of \( a \), is ______.

In a triangle \( \triangle PQR \), the co-ordinates of the points \( P \) and \( Q \) are \( (-2, 4) \) and \( (4, -2) \) respectively. If the equation of the perpendicular bisector of \( PR \) is \( 2x - y + 2 = 0 \), then the centre of the circumcircle of \( \triangle PQR \) is:

The distance between the two points \( A \) and \( A' \) which lie on \( y = 2 \) such that both the line segments \( AB \) and \( A'B \) (where \( B \) is the point \( (2, 3) \)) subtend angle \( \frac{\pi}{4} \) at the origin, is equal to:

Let \( (\alpha,\beta) \) be the centroid of the triangle formed by the lines \( 15x-y=82 \), \( 6x-5y=-4 \) and \( 9x+4y=17 \). Then \( \alpha+2\beta \) and \( 2\alpha-\beta \) are the roots of the equation:

If the orthocentre of the triangle formed by the lines \( 2x + 3y - 1 = 0 \), \( x + 2y - 1 = 0 \) and \( ax + by - 1 = 0 \), is the centroid of another triangle, whose circumcentre and orthocentre respectively are \( (3, 4) \) and \( (-6, -8) \), then the value of \( |a - b| \) is _______.

A variable line \( L \) passes through the point \( (3,5) \) and intersects the positive coordinate axes at the points \( A \) and \( B \). The minimum area of the triangle \( OAB \), where \( O \) is the origin, is:

The equations of the sides \( AB \), \( BC \) and \( CA \) of a triangle \( ABC \) are \( 2x+y=0 \), \( x+py=15a \) and \( x-y=3 \) respectively. If its orthocentre is \( (2,a) \), \( -\dfrac{1}{2}<a<2\), then \(p\) is equal to________.

Let \( A \) be a fixed point \( (0, 6) \) and \( B \) be a moving point \( (2t, 0) \). Let \( M \) be the mid-point of \( AB \) and the perpendicular bisector of \( AB \) meets the \( y \)-axis at \( C \). The locus of the mid-point \( P \) of \( MC \) is:

A line passing through the point \( A(9, 0) \), makes an angle of \( 30^{\circ} \) with the positive direction of \( x \)-axis. If this line is rotated about \( A \) through an angle of \( 15^{\circ} \) in the clockwise direction, then its equation in the new position is:

Let \( C(\alpha, \beta) \) be the circumcenter of the triangle formed by the lines \( 4x + 3y = 69 \), \( 4y - 3x = 17 \) and \( x + 7y = 61 \). Then \( (\alpha - \beta)^2 + \alpha + \beta \) is equal to:

Let \( A\!\left(\dfrac{3}{\sqrt{a}}, \sqrt{a}\right) \), \( a>0 \), be a fixed point in the \( xy \)-plane. The image of \( A \) in \( y \)-axis be \( B \) and the image of \( B \) in \( x \)-axis be \( C \). If \( D(3\cos \theta, a \sin \theta) \) is a point in the fourth quadrant such that the maximum area of \( \triangle ACD \) is \( 12 \) square units, then \( a \) is equal to _________.

If the line \( \ell_{1}: 3y-2x=3 \) is the angular bisector of the lines \( \ell_{2}: x-y+1=0 \) and \( \ell_{3}: \alpha x+\beta y+17=0 \), then \( \alpha^{2}+\beta^{2}-\alpha-\beta \) is equal to ______.

The vertices of a triangle are \( A(-1,3) \), \( B(-2,2) \) and \( C(3,-1) \). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:

In an equilateral triangle \( PQR \), let the vertex \( P \) be at \( (3, 5) \) and the side \( QR \) be along the line \( x + y = 4 \). If the orthocentre of the triangle \( PQR \) is \( (\alpha, \beta) \), then \( 9(\alpha + \beta) \) is equal to:

A light ray emits from the origin making an angle \( 30^\circ \) with the positive \( x \)-axis. After getting reflected by the line \( x+y=1 \), if this ray intersects \( x \)-axis at \( Q \), then the abscissa of \( Q \) is:

Let \( \triangle ABC \) be a triangle formed by the lines \( 7x-6y+3=0 \), \( x+2y-31=0 \) and \( 9x-2y-19=0 \). Let the point \( (h,k) \) be the image of the centroid of \( \triangle ABC \) in the line \( 3x+6y-53=0 \). Then \( h^2+k^2+hk \) is equal to:

Two sides of a parallelogram are along the line \( 4x+5y=0 \) and \( 7x+2y=0 \). If the equation of one of the diagonals of the parallelogram is \( 11x+7y=9 \), then other diagonal passes through the point:

Let \( PQR \) be a triangle. The points \( A, B \) and \( C \) are on the sides \( QR, RP \) and \( PQ \) respectively such that \( \frac{QA}{AR}=\frac{RB}{BP}=\frac{PC}{CQ}=\frac{1}{2} \). Then \( \frac{\text{Area}(\triangle PQR)}{\text{Area}(\triangle ABC)} \) is equal to:

Among the statements:
(S1): If \( A(5,-1) \) and \( B(-2,3) \) are two vertices of a triangle, whose orthocentre is \( (0,0) \), then its third vertex is \( (-4,-7) \)
and
(S2): If positive numbers \( 2a,b,c \) are three consecutive terms of an A.P., then the lines \( ax+by+c=0 \) are concurrent at \( (2,-2) \),

Let the lines \( 3x - 4y - \alpha = 0 \), \( 8x - 11y - 33 = 0 \) and \( 2x - 3y + \lambda = 0 \) be concurrent. If the image of the point \( (1, 2) \) in the line \( 2x - 3y + \lambda = 0 \) is \( \left( \frac{57}{13}, \frac{-40}{13} \right) \), then \( |\alpha \lambda| \) is equal to:

Let the area of a \( \triangle PQR \) with vertices \( P(5, 4) \), \( Q(-2, 4) \) and \( R(a, b) \) be \( 35 \) square units. If its orthocenter and centroid are \( O\!\left(2, \dfrac{14}{5}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to:

If \( (\alpha,\beta) \) is the orthocenter of the triangle \( ABC \) with vertices \( A(3,-7) \), \( B(-1,2) \) and \( C(4,5) \), then \( 9\alpha-6\beta+60 \) is equal to:

The number of integral values of \( m \) so that the abscissa of point of intersection of lines \( 3x + 4y = 9 \) and \( y = mx + 1 \) is also an integer, is:

Let \( A(\alpha,-2), B(\alpha,6) \) and \( C\!\left(\dfrac{\alpha}{4},-2\right) \) be vertices of a \( \triangle ABC \). If \( \left(5,\dfrac{\alpha}{4}\right) \) is the circumcentre of \( \triangle ABC \), then which of the following is NOT correct about \( \triangle ABC \)?

If \( p \) and \( q \) are the lengths of the perpendiculars from the origin on the lines, \( x cosec \alpha - y \,\text{sec}\, \alpha = k \cot 2\alpha \) and \( x \sin \alpha + y \cos \alpha = k \sin 2\alpha \) respectively, then \( k^{2} \) is equal to:

Let the vertex \( A \) of a triangle \( ABC \) be \( (1, 2) \), and the mid-point of the side \( AB \) be \( (5, -1) \). If the centroid of this triangle is \( (3, 4) \) and its circumcenter is \( (\alpha, \beta) \), then \( 21(\alpha + \beta) \) is equal to:

Let \( m_1,m_2 \) be the slopes of two adjacent sides of a square of side \( a \) such that \( a^2+11a+3\!\left(m_1^2+m_2^2\right)=220 \). If one vertex of the square is \( \left(10(\cos\alpha-\sin\alpha),\,10(\sin\alpha+\cos\alpha)\right) \), where \( \alpha\in\!\left(0,\dfrac{\pi}{2}\right) \) and the equation of one diagonal is \( (\cos\alpha-\sin\alpha)x+(\sin\alpha+\cos\alpha)y=10 \), then \( 72(\sin^4\alpha+\cos^4\alpha)+a^2-3a+13 \) is equal to:

Let the angles made with the positive \( x \)-axis by two straight lines drawn from the point \( P(2,3) \) and meeting the line \( x+y=6 \) at a distance \( \sqrt{\frac{2}{3}} \) from the point \( P \) be \( \theta_1 \) and \( \theta_2 \). Then the value of \( (\theta_1+\theta_2) \) is:

Let two straight lines drawn from the origin \( O \) intersect the line \( 3x + 4y = 12 \) at the point \( P \) and \( Q \) such that \( \triangle OPQ \) is an isosceles triangle and \( \angle POQ = 90^\circ \). If \( l = OP^2 + PQ^2 + OQ^2 \), then the greatest integer less than or equal to \( l \) is:

A rod of length \( \text {eight} \, \text{units} \) moves such that its ends \( A \) and \( B \) always lie on the line \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2:1 \) is \( 9( x^{2} + \alpha y^{2} + \beta xy + \gamma x + 28y) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to:

Let \( A(1,0) \), \( B(2,-1) \) and \( C\!\left(\frac{7}{3}, \frac{4}{3}\right) \) be three points. If the equation of the bisector of the angle \( ABC \) is \( \alpha x + \beta y = 5 \), then the value of \( \alpha^2 + \beta^2 \) is:

A ray of light coming from the point \( P(1, 2) \) gets reflected from the point \( Q \) on the \( x \)-axis and then passes through the point \( R(4, 3) \). If the point \( S(h, k) \) is such that \( PQRS \) is a parallelogram, then \( hk^2 \) is equal to:

Let a point \( A \) lie between the parallel lines \( L_1 \) and \( L_2 \) such that its distances from \( L_1 \) and \( L_2 \) are \( 6 \) and \( 3 \) units, respectively. Then the area (in sq. units) of the equilateral triangle \( ABC \), where the points \( B \) and \( C \) lie on the lines \( L_1 \) and \( L_2 \), respectively, is:

Let the equation of the pair of lines, \( y = px \) and \( y = qx \), can be written as \( (y - px)(y - qx) = 0 \). Then the equation of the pair of the angle bisectors of the lines \( x^2 - 4xy - 5y^2 = 0 \) is:

Let \( \alpha, \beta, \gamma, \delta \in \mathbb{Z} \) and let \( A(\alpha,\beta) \), \( B(1,0) \), \( C(\gamma,\delta) \) and \( D(1,2) \) be the vertices of a parallelogram \( ABCD \). If \( AB=\sqrt{10} \) and the points \( A \) and \( C \) lie on the line \( 3y=2x+1 \), then \( 2(\alpha+\beta+\gamma+\delta) \) is equal to:

The portion of the line \( 4x + 5y = 20 \) in the first quadrant is trisected by the lines \( L_1 \) and \( L_2 \) passing through the origin. The tangent of an angle between the lines \( L_1 \) and \( L_2 \) is:

Let \( \triangle ABC \) be an isosceles triangle in which \( A \) is at \( (-1, 0) \), \( \angle A = \dfrac{2\pi}{3} \), \( AB = AC \) and \( B \) is on the positive \( x \)-axis. If \( BC = 4\sqrt{3} \) and the line \( BC \) intersects the line \( y = x + 3 \) at \( (\alpha, \beta) \) then \( \dfrac{\beta^{4}}{\alpha^{2}} \) is __________.

Let a circle of radius \( 4 \) pass through the origin \( O \), the point \( A(-\sqrt{3}\,a, 0) \) and \( B(0, -\sqrt{2}\,b) \), where \( a \) and \( b \) are real parameters and \( ab \neq 0 \). Then the locus of the centroid of \( \triangle OAB \) is a circle of radius:

Let \( A(-1,1),\, B(3,4) \) and \( C(2,0) \) be given three points. A line \( y=mx,\, m>0 \), intersects lines \( AC \) and \( BC \) at point \( P \) and \( Q \) respectively. Let \( A_{1} \) and \( A_{2} \) be the area of \( \triangle ABC \) and \( \triangle PQC \) respectively, such that \( A_{1}=3A_{2} \), then the value of \( m \) is equal to:

A line passing through the point \( P(a, 0) \) makes an acute angle \( \alpha \) with the positive \( x \)-axis. Let this line be rotated about the point \( P \) through an angle \( \dfrac{\alpha}{2} \) in the clock-wise direction. If in the new position, the slope of the line is \( 2 - \sqrt{3} \) and its distance from the origin is \( \dfrac{1}{\sqrt{2}} \), then the value of \( 3a^2 \tan^2 \alpha - 2\sqrt{3} \) is:

Let the line \( L_1 : x + 3 = 0 \) intersect the lines \( L_2 : x - y = 0 \) and \( L_3 : 3x + y = 0 \) at the points \( A \) and \( B \), respectively. Let the bisector of the obtuse angle between the lines \( L_2 \) and \( L_3 \) intersect the line \( L_1 \) at the point \( C \). Then \( BC^2 : AC^2 \) is equal to:

A line, with the slope greater than one, passes through the point \( A(4,3) \) and intersects the line \( x-y-2=0 \) at the point \( B \). If the length of the line segment \( AB \) is \( \dfrac{\sqrt{29}}{3} \), then \( B \) also lies on the line:

Let the point \( P(\alpha,\beta) \) be at a unit distance from each of the two lines \( L_1: 3x-4y+12=0 \), and \( L_2: 8x+6y+11=0 \). If \( P \) lies below \( L_1 \) and above \( L_2 \), then \( 100(\alpha+\beta) \) is equal to:

Let \( B \) and \( C \) be the two points on the line \( y + x = 0 \) such that \( B \) and \( C \) are symmetric with respect to the origin. Suppose \( A \) is a point on \( y - 2x = 2 \) such that \( \triangle ABC \) is an equilateral triangle. Then, the area of the \( \triangle ABC \) is:

A ray of light passing through the point \( P(2, 3) \) reflects on the \( x \)-axis at point \( A \) and the reflected ray passes through the point \( Q(5, 4) \). Let \( R \) be the point that divides the line segment \( AQ \) internally into the ratio \( 2\!:\!1 \). Let the co-ordinates of the foot of the perpendicular \( M \) from \( R \) on the bisector of the angle \( PAQ \) be \( (\alpha, \beta) \). Then, the value of \( 7\alpha + 3\beta \) is equal to ________.

Let \( A(1,2) \) and \( C(-3,-6) \) be two diagonally opposite vertices of a rhombus, whose sides \( AD \) and \( BC \) are parallel to the line \( 7x - y = 14 \). If \( B(\alpha,\beta) \) and \( D(\gamma,\delta) \) are the other two vertices, then \( |\alpha + \beta + \gamma + \delta| \) is equal to:

Let \( A(-1,1) \) and \( B(2,3) \) be two points and \( P \) be a variable point above the line \( AB \) such that the area of \( \triangle PAB \) is \( 10 \). If the locus of \( P \) is \( ax+by=15 \), then \( 5a+2b \) is:

If the image of the point \( (-4,5) \) in the line \( x+2y=2 \) lies on the circle \( (x+4)^{2}+(y-3)^{2}=r^{2} \), then \( r \) is equal to:

Let \( a \) be the length of a side of a square \( OABC \) with \( O \) being the origin. Its side \( OA \) makes an acute angle \( \alpha \) with the positive \( x \)-axis and the equations of its diagonals are \( (\sqrt{3}+1)x + (\sqrt{3}-1)y = 0 \) and \( (\sqrt{3}-1)x - (\sqrt{3}+1)y + 8\sqrt{3} = 0 \). Then \( a^{2} \) is equal to:

Let \( A \) be the set of all points \( (\alpha,\,\beta) \) such that the area of triangle formed by the points \( (5,\,6) \), \( (3,\,2) \) and \( (\alpha,\,\beta) \) is \( 12 \) square units. Then the least possible length of a line segment joining the origin to a point in \( A \), is:

Let a triangle be bounded by the lines \( L_1: 2x + 5y = 10 \); \( L_2: -4x + 3y = 12 \) and the line \( L_3 \), which passes through the point \( P(2,3) \), intersects \( L_2 \) at \( A \) and \( L_1 \) at \( B \). If the point \( P \) divides the line-segment \( AB \), internally in the ratio \( 1:3 \), then the area of the triangle is equal to:

Let the mid points of the sides of a triangle \( ABC \) be \( \left(\dfrac{5}{2},7\right) \), \( \left(\dfrac{5}{2},3\right) \) and \( (4,5) \). If its incentre is \( (h,k) \), then \( 3h+k \) is equal to:

Let the area of the triangle formed by a straight line \( L: x + by + c = 0 \) with co-ordinate axes be \( 48 \) square units. If the perpendicular drawn from the origin to the line \( L \) makes an angle of \( 45^\circ \) with the positive \( x \)-axis, then the value of \( b^2 + c^2 \) is:

The point \( P(a, b) \) undergoes the following three transformations successively:
(a) Reflection about the line \( y = x \)
(b) Translation through \( 2 \) units along the positive direction of x-axis
(c) Rotation through angle \( \dfrac{\pi}{4} \) about the origin in the anti-clockwise direction
If the co-ordinates of the final position of the point \( P \) are \( \left( -\dfrac{1}{\sqrt{2}}, \dfrac{7}{\sqrt{2}} \right) \), then the value of \( 2a + b \) is equal to:

Let \( ABC \) be an equilateral triangle with orthocenter at the origin and the side \( BC \) on the line \( x + 2\sqrt{2}\,y = 4 \). If the co-ordinates of the vertex \( A \) are \( (\alpha, \beta) \) then the greatest integer less than or equal to \( \lvert \alpha + \sqrt{2}\beta \rvert \) is:

The equations of the sides \( AB \) and \( AC \) of a triangle \( ABC \) are \( (\lambda + 1)x + \lambda y = 4 \) and \( \lambda x + (1-\lambda)y + \lambda = 0 \) respectively. Its vertex \( A \) is on the y-axis and its orthocentre is \( (1, 2) \). The length of the tangent from the point \( C \) to the part of the parabola \( y^{2} = 6x \) in the first quadrant is:

Let \( \tan\alpha, \tan\beta \) and \( \tan\gamma \,;\ \alpha,\beta,\gamma \neq \dfrac{(2n-1)\pi}{2},\ n\in \mathbb{N} \) be the slopes of three line segments \( OA \), \( OB \) and \( OC \), respectively, where \( O \) is origin. If circumcentre of \( \triangle ABC \) coincides with origin and its orthocentre lies on \( y \)-axis, then the value of \( \left(\dfrac{\cos 3\alpha+\cos 3\beta+\cos 3\gamma}{\cos\alpha\cos\beta\cos\gamma}\right)^{2} \) is equal to ________.

The equations of two sides \( AB \) and \( AC \) of a triangle \( ABC \) are \( 4x + y = 14 \) and \( 3x - 2y = 5 \), respectively. The point \( \left(2, \frac{4}{3}\right) \) divides the third side \( BC \) internally in the ratio \( 2:1 \), the equation of the side \( BC \) is:

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines \( L_1: 2x + y + 6 = 0 \) and \( L_2: 4x + 2y - p = 0, \, p > 0 \), at the points \( A \) and \( B \), respectively. If \( AB = \frac{9}{\sqrt{2}} \) and the foot of the perpendicular from the point \( A \) on the line \( L_2 \) is \( M \), then \( \frac{AM}{BM} \) is equal to:

Let \( A(a,b) \), \( B(3,4) \) and \( C(-6,-8) \) respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point \( P(2a+3,7b+5) \) from the line \( 2x+3y-4=0 \) measured parallel to the line \( x-2y-1=0 \) is:

Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \(\displaystyle \frac{4}{9} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is:

Let \( R \) be the interior region between the lines \( 3x-y+1=0 \) and \( x+2y-5=0 \) containing the origin. The set of all values of \( a \), for which the points \( (a^2,a+1) \) lie in \( R \), is:

Let the equation of two adjacent sides of a parallelogram \( ABCD \) be \( 2x - 3y = -23 \) and \( 5x + 4y = 23 \). If the equation of its one diagonal \( AC \) is \( 3x + 7y = 23 \) and the distance of \( A \) from the other diagonal is \( d \), then \( 50\,d^2 \) is equal to \( \_\_\_\_\_ \).

The distance of the point \( (2,3) \) from the line \( 2x-3y+28=0 \), measured parallel to the line \( \sqrt{3}x-y+1=0 \), is equal to:

Consider a triangle having vertices \( A(-2,3) \), \( B(1,9) \) and \( C(3,8) \). If a line \( L \) passing through the circum-center of triangle \( ABC \), bisects line \( BC \), and intersects y-axis at point \( \left(0,\dfrac{\alpha}{2}\right) \), then the value of real number \( \alpha \) is ________.

If the orthocenter of the triangle formed by the lines \( y=x+1 \), \( y=4x-8 \) and \( y=mx+c \) is at \( (3,-1) \), then \( m-c \) is:

Let the points of intersections of the lines \( x - y + 1 = 0 \), \( x - 2y + 3 = 0 \) and \( 2x - 5y + 11 = 0 \) are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is:

Let the centroid of an equilateral triangle \( ABC \) be at the origin. Let one of the sides of the equilateral triangle be along the straight line \( x + y = 3 \). If \( R \) and \( r \) be the radius of circumcircle and incircle respectively of \( \triangle ABC \), then \( (R + r) \) is equal to:

Let the points \( \left(\frac{11}{2}, \alpha\right) \) lie on or inside the triangle with sides \( x + y = 11 \), \( x + 2y = 16 \) and \( 2x + 3y = 29 \). Then the product of the smallest and the largest values of \( \alpha \) is equal to:

Let \( A(-2,-1), B(1,0), C(\alpha ,\beta ) \) and \( D(\gamma ,\delta ) \) be the vertices of a parallelogram \( ABCD \). If the point \( C \) lies on \( 2x-y=5 \) and the point \( D \) lies on \( 3x-2y=6 \), then the value of \( \left|\alpha +\beta +\gamma +\delta \right| \) is equal of ________.

Let \( \left(5, \frac{a}{4}\right) \) be the circumcenter of a triangle with vertices \( A(a, -2) \), \( B(a, 6) \) and \( C\!\left(\frac{a}{4}, -2\right) \). Let \( \alpha \) denote the circumradius, \( \beta \) denote the area and \( \gamma \) denote the perimeter of the triangle. Then \( \alpha + \beta + \gamma \) is:

Let \( C \) be the circle \( x^{2} + (y - 1)^{2} = 2 \), \( E_{1} \) and \( E_{2} \) be two ellipse whose centres lie at the origin and major axes lie on \( x \)-axis and \( y \)-axis respectively. Let the straight line \( x + y = 3 \) touch the curves \( C, E_{1} \) and \( E_{2} \) at \( P(x_{1},y_{1}), Q(x_{2},y_{2}) \) and \( R(x_{3},y_{3}) \) respectively. Given that \( P \) is the mid-point of the line segment \( QR \) and \( PQ = \dfrac{2\sqrt{2}}{3} \), the value of \( 9(x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3}) \) is equal to ______.

In an isosceles triangle \( ABC \), the vertex \( A \) is \( (6,1) \) and the equation of the base \( BC \) is \( 2x + y = 4 \). Let the point \( B \) lie on the line \( x + 3y = 7 \). If \( (\alpha, \beta) \) is the centroid of \( \triangle ABC \), then \( 15(\alpha + \beta) \) is equal to:

Two vertices of a triangle \( ABC \) are \( A(3,-1) \) and \( B(-2,3) \), and its orthocenter is \( P(1,1) \). If the coordinates of the point \( C \) are \( (\alpha,\beta) \) and the centre of the circle circumscribing the triangle \( PAB \) is \( (h,k) \), then the value of \( (\alpha+\beta)+2(h+k) \) equals

Let \( ABC \) be a triangle with \( A(-3,1) \) and \( \angle ACB=\theta \), \( 0<\theta<\frac{\pi}{2} \). If the equation of the median through \( B \) is \( 2x+y-3=0 \) and the equation of angle bisector of \( C \) is \( 7x-4y-1=0 \), then \( \tan\theta \) is equal to:

If a straight line drawn through the point of intersection of the lines \( 4x+3y-1=0 \) and \( 3x+4y-1=0 \), meets the co-ordinate axes at the points \( P \) and \( Q \), then the locus of the mid-point of \( PQ \) is:

A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to:

Two equal sides of an isosceles triangle are along \( -x + 2y = 4 \) and \( x + y = 4 \). If \( m \) is the slope of its third side, then the sum, of all possible distinct values of \( m \), is:

If the line segment joining the point \( (5,2) \) and \( (2,a) \) subtends an angle \( \dfrac{\pi}{4} \) at the origin, then the absolute value of the product of all possible values of \( a \) is:

Consider the lines \( x(3\lambda+1)+y(7\lambda+2)=17\lambda+5 \), \( \lambda \) being a parameter, all passing through a point \( P \). One of these lines (say \( L \)) is farthest from the origin. If the distance of \( L \) from the point \( (3, 6) \) is \( d \), then the value of \( d^2 \) is:

A man starts walking from the point \( P(-3, 4) \), touches the \( x \)-axis at \( R \), and then turns to reach at the point \( Q(0, 2) \). The man is walking at a constant speed. If the man reaches the point \( Q \) in the minimum time, then \( 50\big((PR)^2 + (RQ)^2\big) \) is equal to:

In a \( \triangle ABC \), suppose \( y = x \) is the equation of the bisector of the angle \( B \) and the equation of the side \( AC \) is \( 2x - y = 2 \). If \( 2AB = BC \) and the points \( A \) and \( B \) are respectively \( (4, 6) \) and \( (\alpha, \beta) \), then \( \alpha + 2\beta \) is equal to:

The distance of the origin from the centroid of the triangle whose two sides have the equations \( x - 2y + 1 = 0 \) and \( 2x - y - 1 = 0 \) and whose orthocenter is \( \left(\frac{7}{3}, \frac{7}{3}\right) \) is:

Let a variable line of slope \( m > 0 \) passing through the point \( (4,-9) \) intersect the coordinate axes at the points \( A \) and \( B \). The minimum value of the sum of the distance of \( A \) and \( B \) from the origin is:

Let A be the point of intersection of the lines \( 3x + 2y = 14 \), \( 5x - y = 6 \) and B be the point of intersection of the lines \( 4x + 3y = 8 \), \( 6x + y = 5 \). The distance of the point \( P(5,-2) \) from the line AB is:

The straight lines \( l_1 \) and \( l_2 \) pass through the origin and trisect the line segment of the line \( L: 9x + 5y = 45 \) between the axes. If \( m_1 \) and \( m_2 \) are the slopes of the lines \( l_1 \) and \( l_2 \), then the point of intersection of the line \( y = (m_1 + m_2)x \) with \( L \) lies on:

Let \( A(a,0) \), \( B(b, 2b + 1) \) and \( C(0,b) \), \( b \neq 0 \), \( |b| \neq 1 \), be points such that the area of triangle \( ABC \) is \( 1 \, \text{sq. unit} \), then the sum of all possible values of \( a \) is :

Let the triangle \( PQR \) be the image of the triangle with vertices \( (1, 3) \), \( (3, 1) \) and \( (2, 4) \) in the line \( x + 2y = 2 \). If the centroid of \( \triangle PQR \) is the point \( (\alpha, \beta) \), then \( 15(\alpha - \beta) \) is equal to:

Let \( \triangle ABC \) be the triangle such that the equations of lines \( AB \) and \( AC \) be \( 3y - x = 2 \) and \( x + y = 2 \), respectively, and the points \( B \) and \( C \) lie on \( x \)-axis. If \( P \) is the orthocentre of the triangle \( ABC \), then the area of the triangle \( PBC \) is equal to:

Let the area of the triangle with vertices \( A(1,\alpha) \), \( B(\alpha,0) \) and \( C(0,\alpha) \) be \( 4 \, \text{sq. units} \). If the points \( (\alpha,-\alpha) \), \( (-\alpha,\alpha) \) and \( (\alpha^{2},\beta) \) are collinear, then \( \beta \) is equal to:

If the sum of squares of all real values of \( \alpha \), for which the lines \( 2x - y + 3 = 0 \), \( 6x + 3y + 1 = 0 \) and \( a x + 2y - 2 = 0 \) do not form a triangle is \( p \), then the greatest integer less than or equal to \( p \) is __________.

If the orthocentre of the triangle, whose VERTICES are \( (1,2),(2,3) \) and \( (3,1) \) is \( (\alpha,\beta) \), then the quadratic equation whose roots are \( \alpha+4\beta \) and \( 4\alpha+\beta \), is:

If \( P(6,1) \) be the orthocentre of the triangle whose vertices are \( A(5,-2), B(8,3) \) and \( C(h,k) \), then the point \( C \) lies on the circle:

Let the three sides of a triangle are on the lines $4x - 7y + 10 = 0$, $x + y = 5$ and $7x + 4y = 15$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x = 0$, $y = 0$ and $x + y = 1$ is:

The area of the quadrilateral \( ABCD \) with vertices \( A(2,1,1) \), \( B(1,2,5) \), \( C(-2,-3,5) \) and \( D(1,-6,-7) \) is equal to:

Let the circumcentre of a triangle with vertices \( A(a,3) \), \( B(b,5) \) and \( C(a,b) \), \( ab > 0 \) be \( P(1,1) \). If the line \( AP \) intersects the line \( BC \) at the point \( Q(k_1,k_2) \), then \( k_1 + k_2 \) is equal to:

Let \( R \) be the point \( (3,7) \) and let \( P \) and \( Q \) be two points on the line \( x+y=5 \) such that \( PQR \) is an equilateral triangle. Then the area of \( \triangle PQR \) :

Let \( A(6,8) \), \( B(10\cos\alpha,-10\sin\alpha) \) and \( C(-10\sin\alpha,10\cos\alpha) \), be the vertices of a triangle. If \( L(a,9) \) and \( G(h,k) \) be its orthocenter and centroid respectively, then \( (5a-3h+6k+100\sin 2\alpha )\) is equal to ____________.

Let \( A(1, 1), \; B(-4, 3), \; C(-2, -5) \) be vertices of a triangle \( ABC \). \( P \) be a point on side \( BC \), \( \Delta_{1} \) and \( \Delta_{2} \) be the areas of triangles \( APB \) and \( ABC \), respectively. If \( \Delta_{1} : \Delta_{2} = 4 : 7 \), then the area enclosed by the lines \( AP \), \( AC \) and the \( x \)-axis is:

Let a ray of light passing through the point \( (3,10) \) reflects on the line \( 2x+y=6 \) and the reflected ray passes through the point \( (7,2) \). If the equation of the incident ray is \( ax+by+1=0 \), then \( a^2+b^2+3ab \) is equal to ________.

The equation of one of the straight lines which passes through the point \( (1, 3) \) and makes an angle \( \tan^{-1}(\sqrt{2}) \) with the straight line, \( y + 1 = 3\sqrt{2}\,x \) is:

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JEE Main 2023 15th April Morning Shift

If \( (\alpha,\beta) \) is the orthocenter of the triangle \( ABC \) with vertices \( A(3,-7) \), \( B(-1,2) \) and \( C(4,5) \), then \( 9\alpha-6\beta+60 \) is equal to: