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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:

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 \( A\!\left(\dfrac{3}{\sqrt{a}}, \sqrt{a}\right) \), \( a>0 \), 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 _________.

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 \( \triangle ABC \)?

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:

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:

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 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:

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:

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 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:

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 \( A(1, 1), \; B(-4, 3), \; C(-2, -5) \) 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:

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JEE Main 2022 29th June Evening Shift

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: