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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:

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:

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:

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:

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 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 \( A(-1,1),\, B(3,4) \) 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:

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:

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 \( \tan\alpha, \tan\beta \) 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 ________.

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 ________.

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 \( ABC \) 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:

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:

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 :

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 2021 25th July Evening Shift

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: