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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:

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:

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:

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 \( a \) 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 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:

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 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:

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 \( \left(\frac{11}{2}, \alpha\right) \) 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 \( C \) 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 ______.

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:

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:

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 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:

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 ____________.

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JEE MAIN 2025 4th April Morning Shift

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: