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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 ______.

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:

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:

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 ______.

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 \( PQR \) 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:

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:

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:

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 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 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:

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:

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:

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JEE Main 2023 29th Jan Morning Shift

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: