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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:

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:

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:

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 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 \( \alpha, \beta, \gamma, \delta \in \mathbb{Z} \) 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 \( A(-1,1) \) 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:

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:

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 \( R \) 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:

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:

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:

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

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:

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:

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:

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 \( P(6,1) \) 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 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 ________.

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JEE Main 2024 27th Jan Morning Shift

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: