List of top Mathematics Questions on Coordinate Geometry asked in JEE Main

If the image of the point \((-4, 5)\) in the line \(x + 2y = 2\) lies on the circle \((x + 4)^2 + (y - 3)^2 = r^2\), then \(r\) is equal to:

Let a circle passing through (2, 0) have its centre at the point \( (h, k) \). Let \( (x_c, y_c) \) be the point of intersection of the lines \( 3x + 5y = 1 \) and \( (2 + c)x + 5c^2y = 1 \). If \( h = \lim_{c \to 1} x_c \) and \( k = \lim_{c \to 1} y_c \), then the equation of the circle is:

A variable line \( L \) passes through the point \( (3, 5) \) and intersects the positive coordinate axes at the points \( A \) and \( B \). The minimum area of the triangle \( OAB \), where \( O \) is the origin, is:

A ray of light coming from the point \( P(1, 2) \) gets reflected from the point \( Q \) on the x-axis and then passes through the point \( R(4, 3) \). If the point \( S(h, k) \) is such that \( PQRS \) is a parallelogram, then \( hk^2 \) is equal to:

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 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 \( A(-2, -1) \), \( B(1, 0) \), \( C(\alpha, \beta) \), 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\( \_\_\_\_\_\).

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

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 _____

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

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 \( P \) 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

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

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 line passing through the points P (2, –1, 2) and Q (5, 3, 4) meet the plane x – y + z = 4 at the point R. Then the distance of the point R from the plane x + 2y + 3z + 2 = 0 measured parallel to the line \(\frac{x-7}{2}=\frac{y+3}{2}=\frac{z-2}{1}\) is equal to

Let α be the area of the larger region bounded by the curve y² = 8x and the lines y = x and x = 2, which lies in the first quadrant. Then the value of 3α is equal to:

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

Area bounded by larger part in I quadrant by x = 4y² , x = 2 and y = x is A then 3A equals

A straight line cuts off the intercepts OA = a and OB = b on the positive directions of the x-axis and y-axis, respectively. If the perpendicular from the origin O to this line makes an angle of \(\frac{\pi}{6}\) with the positive direction of the y-axis and the area of △OAB is \(\frac{98\sqrt{3}}{3},\) then a² − b² is equal to:

The shortest distance between the lines $\frac{x−1}2 =\frac{y−2}1 =\frac{z−6}{-3} $ and $\frac{x−1}2 =\frac{y+8}{-7} =\frac{z−4}{5} $ 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 : 

Let P be the plane passing through the points (5, 3, 0), (13, 3, –2) and (1, 6, 2). For α ∈ N, if the distances of the points A (3, 4, α) and B (2, α, a) from the plane P are 2 and 3 respectively, then the positive value of a is