List of top Mathematics Questions on Coordinate Geometry

Suppose AB is a focal chord of the parabola \( y^2 = 12x \) of length \( l \) and slope \( m<\sqrt{3} \). If the distance of the chord AB from the origin is \( d \), then \( ld^2 \) is equal to _________.

Let two straight lines drawn from the origin \( O \) intersect the line \(3x + 4y = 12\) at the points \( P \) and \( Q \) such that \( \triangle OPQ \) is an isosceles triangle and \( \angle POQ = 90^\circ \).  If \( l = OP^2 + PQ^2 + QO^2 \), then the greatest integer less than or equal to \( l \) is:

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a + b)² is equal to :

Let the foci and length of the latus rectum of an ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b \quad \text{be } (\pm 5, 0) \text{ and } \sqrt{50}, \] respectively. Then, the square of the eccentricity of the hyperbola \[ \frac{x^2}{b^2} - \frac{y^2}{a^2 b^2} = 1 \] equals _____

Let \( \alpha, \beta, \gamma, \delta \in \mathbb{Z} \) and let \( A (\alpha, \beta) \), \( B (1, 0) \), \( C (\gamma, \delta) \), and \( D (1, 2) \) be the vertices of a parallelogram \( ABCD \). If \( AB = \sqrt{10} \) and the points \( A \) and \( C \) lie on the line \( 3y = 2x + 1 \), then \( 2 (\alpha + \beta + \gamma + \delta) \) is equal to

If one of the diameters of the circle \( x^2 + y^2 - 10x + 4y + 13 = 0 \) is a chord of another circle \( C \), whose center is the point of intersection of the lines \( 2x + 3y = 12 \) and \( 3x - 2y = 5 \),then the radius of the circle \( C \) is

If the foci of a hyperbola are the same as that of the ellipse \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \) and the eccentricity of the hyperbola is \( \frac{15}{8} \) times the eccentricity of the ellipse,then the smaller focal distance of the point \( \left( \sqrt{2}, \frac{14}{3} \sqrt{5} \right) \) on the hyperbola, is equal to

Let \( A(-2, -1) \), \( B(1, 0) \), \( C(\alpha, \beta) \), and \( D(\gamma, \delta) \) be the vertices of a parallelogram \( ABCD \). If the point \( C \) lies on \( 2x - y = 5 \) and the point \( D \) lies on \( 3x - 2y = 6 \), then the value of \( | \alpha + \beta + \gamma + \delta | \) is equal to\( \_\_\_\_\_\).

If the function \( f : (-\infty, -1] \rightarrow (a, b) \) defined by \(f(x) = e^{x^3 - 3x + 1}\) is one-one and onto, then the distance of the point \( P(2b + 4, a + 2) \) from the line \( x + e^{-3} y = 4 \) is:

The area of the region enclosed by the parabola \( y = 4x - x^2 \) and \( 3y = (x - 4)^2 \) is equal to

Let \( P \) be a parabola with vertex \( (2, 3) \) and directrix \( 2x + y = 6 \). Let an ellipse \( E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \), of eccentricity \( \frac{1}{\sqrt{2}} \) pass through the focus of the parabola \( P \). Then the square of the length of the latus rectum of \( E \) is

Let a variable line passing through the centre of the circle x² + y² – 16x – 4y = 0, meet the positive co-ordinate axes at the point A and B. Then the minimum value of OA + OB, where O is the origin, is equal to

Let \( A (a, b), B(3, 4) \) and \( (-6, -8) \) respectively denote the centroid, circumcentre, and orthocentre of a triangle. Then, the distance of the point \( P(2a + 3, 7b + 5) \) from the line \( 2x + 3y - 4 = 0 \) measured parallel to the line \( x - 2y - 1 = 0 \) is

Two vertices of a triangle \( \triangle ABC \) are \( A(3, -1) \) and \( B(-2, 3) \), and its orthocentre is \( P(1, 1) \). If the coordinates of the point \( C \) are \( (\alpha, \beta) \) and the centre of the circle circumscribing the triangle \( \triangle PAB \) is \( (h, k) \), then the value of \[ (\alpha + \beta) + 2(h + k) \] equals:

Consider a circle \( (x - \alpha)^2 + (y - \beta)^2 = 50 \), where \( \alpha, \beta> 0 \). If the circle touches the line \( y + x = 0 \) at the point \( P \), whose distance from the origin is \( 4\sqrt{2} \), then \( (\alpha + \beta)^2 \) is equal to ....

Let \( R \) be the interior region between the lines \( 3x - y + 1 = 0 \) and \( x + 2y - 5 = 0 \) containing the origin. The set of all values of \( a \), for which the points \( (a^2, a + 1) \) lie in \( R \), is:

Let the image of the point \( (1, 0, 7) \) in the line \[ \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \] be the point \( (\alpha, \beta, \gamma) \). Then which one of the following points lies on the line passing through \( (\alpha, \beta, \gamma) \) and making angles \( \frac{2\pi}{3} \) and \( \frac{3\pi}{4} \) with the y-axis and z-axis respectively and an acute angle with the x-axis?

Consider two circles \( C_1 : x^2 + y^2 = 25 \) and \( C_2 : (x - \alpha)^2 + y^2 = 16 \), where \( \alpha \in (5, 9) \). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of \( C_1 \) and \( C_2 \) be\[\sin^{-1} \left( \frac{\sqrt{63}}{8} \right).\]If the length of the common chord of \( C_1 \) and \( C_2 \) is \( \beta \), then the value of \( (\alpha \beta)^2 \) equals ____.

Let \( A(10, 0) \) and \( B(0, \beta) \) be the points on the line \( 5x + 7y = 50 \). Let the point \( P \) divide the line segment \( AB \) internally in the ratio \( 7 : 3 \). Let \( 3x - 25 = 0 \) be a directrix of the ellipse \( E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and the corresponding focus be \( S \). If from \( S \), the perpendicular on the x-axis passes through \( P \), then the length of the latus rectum of \( E \) is equal to

Let \( P(\alpha, \beta) \) be a point on the parabola \( y^2 = 4x \). If \( P \) also lies on the chord of the parabola \( x^2 = 8y \) whose midpoint is \( \left( 1, \frac{5}{4} \right) \), then \( (\alpha - 28)(\beta - 8) \) is equal to ______.

If the mean and variance of five observations are \( \frac{24}{5} \) and \( \frac{194}{25} \) respectively and the mean of first four observations is \( \frac{7}{2} \), then the variance of the first four observations is equal to

Let \( P \) be a point on the hyperbola \( H: \frac{x^2}{9} - \frac{y^2}{4} = 1 \), in the first quadrant such that the area of the triangle formed by \( P \) and the two foci of \( H \) is \( 2 \sqrt{13} \). Then, the square of the distance of \( P \) from the origin is

If the lines \(\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}\)  and \(\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}\) intersect at the point 𝑃, then the distance of the point 𝑃 from the plane 𝑧=𝑎 is : 

In the chessboard if the rectangle form is 1296, find the total number of squares formed.

The plane \(2x−y+z=4\) intersects the line segment joining the points \(A(a,−2,4)\) and \(B(2,b,−3)\) at the point C in the ratio \(2:1\) and the distance of the point C from the origin is \(\sqrt5\). If \(ab<0 \) and \(P\) is the point \((a−b,b,2b−a)\). Then \(CP^2 \) is equal to