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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 1st Feb Morning Shift

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: