List of top Mathematics Questions

The centroid of a triangle with vertices \( (a, 1, 3), (-2, b, -5), (4, 7, c) \) lies at the origin. Find \( a^2 + b^2 + c^2 \).
The equations of the lines perpendicular to \(x^2 - 5xy + 4y^2 = 0\) and passing through (2, 1) is
The point on the curve $ y = x^2 + 4x + 3 $ that is closest to the line $ y = 3x + 2 $ is:
The length of the latus rectum of the parabola \( y^2-4x-4y+8=0 \) is
If \( \left(x_1, \frac{1}{x_1}\right), \left(x_2, \frac{1}{x_2}\right), \left(x_3, \frac{1}{x_3}\right), \left(x_4, \frac{1}{x_4}\right) \) lie on the boundary of a circle of radius 4, then the value of \( x_1 x_2 x_3 x_4 \) is:
A stick of length $r$ units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is
If the equation of the tangent to the circle \(x^2+y^2=a^2\) at \(P(x_1, y_1)\) is \(x x_1 + y y_1 = K\), then K =
For $\alpha \in \left[0, \dfrac{\pi}{2} \right]$, the angle between the lines represented by
$[x \cos\theta - y] \left[ (\cos\theta + \tan\alpha)x - (1 - \cos\theta \tan\alpha)y \right] = 0$ is
Suppose $P$ and $Q$ are the midpoints of the sides $AB$ and $BC$ of a triangle where $A(1, 3)$, $B(3, 7)$ and $C(7, 15)$ are vertices. Then the locus of $R$ satisfying $AC^2 + QR^2 = PR^2$ is
Find the locus of midpoints of points of intersection of the line $$ x \cos \theta + y \sin \theta = 1 $$ with the coordinate axes.
Suppose ABOC is a rhombus in the first quadrant with O being the origin. If the vertices B and C of \(\triangle ABC\) lie respectively on \(y=\frac{x}{\sqrt{3}}\) and \(y=0\), and the side BC passes through \((\frac{2}{3}, \frac{2}{3})\), then the midpoint of BC is
Points $ P $ and $ Q $ lie on the lines: $$ 3x + 4y - 4 = 0 \quad \text{and} \quad 5x - y - 4 = 0 $$ The midpoint of segment $ PQ $ is $ (1, 5) $. Find the slope of the line $ PQ $.
Suppose a point \( P \) moves such that: \[ BP^2 - AP^2 = 121 \] where \( A = (2, 5) \), \( B = (5, 11) \). Then the locus of \( P \) is a straight line. What is the slope of this line?
Sides passing through vertex $ (\alpha, \beta) $ of a triangle are bisected at right angles by the lines: $$ y^2 - 8xy - ax^2 = 0 $$ Find the coordinates of the centroid of the triangle.
A point P(x, y) is such that the sum of squares of its distances from (a, 0) and (--a, 0) is \(2b^2\). The equation representing the locus of P is
For any real number $t$, the point $\left( \dfrac{8t}{1 + t^2}, \dfrac{4(1 - t^2)}{1 + t^2} \right)$ lies on a/an
Suppose $\triangle ABC$ is an isosceles triangle with $\angle C = 90^\circ$, $A = (2, 3)$ and $B = (4, 5)$. Then the centroid of the triangle is
Find the locus of the incenter of the triangle formed by: \[ xy - 4x - 4y + 16 = 0,\quad x + y = 5 \]
The equation of bisectors of the angle between the lines given by \( 3x^2 + 5xy + 4y^2 = 0 \) is:
If one side of an isosceles triangle is given by \( y = 2 \), and the base is provided by the points \( (2, 0) \) and \( (0, 2) \), then its area (in sq. units) is:
If the points of intersection of the coordinate axes and $|x + y| = 2$ form a rhombus, then its area is
A circle passes through \( (0, a) \) and \( (b, h) \), and its center is at \( (c, 0) \). Find \( c \).
The area of the circle passing through the points $(5, \pm2)$, $(1,2)$ is
The equation of the tangent to the circle \( x^2 + y^2 = 9 \) making an angle \( 60^\circ \) with the x-axis is:
If the point \( P(\sin\alpha, \cos\alpha) \) lies inside the triangle formed by the vertices \( (0, 0), \left(\frac{\sqrt{3}}{2}, 0\right), (0, \frac{\sqrt{3}}{2}) \), then \( \alpha \) lies in the interval: