List of top Mathematics Questions

The equation of the pair of transverse common tangents drawn to the circles \( x^2 + y^2 + 2x + 2y + 1 = 0 \) and \( x^2 + y^2 - 2x - 2y + 1 = 0 \) is:
If \( (1,1), (-2,2), (2,-2) \) are 3 points on a circle \( S \), then the perpendicular distance from the center of the circle \( S \) to the line \( 3x - 4y + 1 = 0 \) is:
A circle passing through the points \( (1, 1) \) and \( (2, 0) \) touches the line \( 3x - y - 1 = 0 \). If the equation of this circle is \( x^2 + y^2 + 2gx + 2fy + c = 0 \), then a possible value of \( g \) is

In a triangle \(ABC\), \(\displaystyle \frac{a(rr_1+r_2r_3)}{r_1-r+r_2r_3} =\;?\)

Area of the triangle bounded by the lines given by the equations: \[ 12x^2 - 20xy + 7y^2 = 0 \quad \text{and} \quad x + y - 1 = 0 \]
If the equation of the transverse common tangent of the circles \(x^2 + y^2 - 4x + 6y + 4 = 0\) and \(x^2 + y^2 + 2x - 2y - 2 = 0\) is \(ax + by + c = 0\), then \(\frac{a}{c} =\)
The curve \(y = x^3 - 2x^2 + 3x - 4\) intersects the horizontal line \(y = -2\) at the point \(P(h,k)\). If the tangent drawn to this curve at \(P\) meets the X-axis at \((x_1, y_1)\), then \(x_1 =\)
The power of a point \( (2,0) \) with respect to a circle \( S \) is \(-4\) and the length of the tangent drawn from the point \( (1,1) \) to \( S \) is \( 2 \). If the circle \( S \) passes through the point \( (-1,-1) \), then the radius of the circle \( S \) is:
A(1, -2, 1) and B(2, -1, 2) are the end points of a line segment. If \(D(\alpha, \beta, \gamma)\) is the foot of the perpendicular drawn from \(C(1, 2, 3)\) to AB, then \(\alpha^2 + \beta^2 + \gamma^2 =\)
A, B, C are the vertices of a triangle ABC. If the bisector of \( \angle BAC \) intersects the side BC at D\((p,q,r)\), then \( \sqrt{2p+q+r} = ? \)
If the line \(L = x \cos \alpha + y \sin \alpha - p = 0\) represents a line perpendicular to the line \(x + y + 1 = 0\) and \(p\) is positive, \(a\) lies in the fourth quadrant and perpendicular distance from \(\left(\sqrt{2}, \sqrt{2}\right)\) to the line \(L = 0\) is 5 units, then find \(p\):
The vertices of triangle \( \Delta ABC \) are \( A(2, 3, k) \), \( B(-1, k, -1) \), and \( C(4, -3, 2) \). If \( AB = AC \) and \( k>0 \), then the triangle \( ABC \) is:
If \( (l, k) \) is a point on the circle passing through the points \( (-1, 1) \), \( (0, -1) \), and \( (1, 0) \), and if \( k \neq 0 \), then find \( k \).
If \( A(1,2) \), \( B(2,1) \) are two vertices of an acute angled triangle and \( S(0,0) \) is its circumcenter, then the angle subtended by \( AB \) at the third vertex is
The line \( x + y + 1 = 0 \) intersects the circle \( x^2 + y^2 - 4x + 2y - 4 = 0 \) at the points A and B. If \( M(a, b) \) is the midpoint of AB, then \( a - b = \)?
(0, k) is the point to which the origin is to be shifted by the translation of the axes so as to remove the first degree terms from the equation \( ax^2 - 2xy + by^2 - 2x + 4y + 1 = 0 \) and \( \frac{1}{2} \tan^{-1}(2) \) is the angle through which the coordinate axes are to be rotated about the origin to remove the \( xy \)-term from the given equation, then \( a + b = \):
The slope of a common tangent to the circles \( x^2 + y^2 - 4x - 8y + 16 = 0 \) and \( x^2 + y^2 - 6x - 16y + 64 = 0 \) is:
The line \( 2x + y - 3 = 0 \) divides the line segment joining the points \( A(1,2) \) and \( B(2,-1) \) in the ratio \( a:b \) at the point \( C \). If the point \( C \) divides the line segment joining the points \( P\left( \frac{b}{3a}, -3 \right) \) and \( Q\left( -3, \frac{-b}{3a} \right) \) in the ratio \( p:q \), then \( \frac{p}{q} + \frac{q}{p} = \):
If \( (p, q) \) is the center of the circle which cuts the three circles \( x^2 + y^2 - 2x - 4y + 4 = 0 \), \( x^2 + y^2 + 2x - 4y + 1 = 0 \), and \( x^2 + y^2 - 4x - 2y - 11 = 0 \) orthogonally, then find \( p + q \).
The slope of one of the pair of lines \(2x^2 + hxy + 6y^2 = 0\) is three times the slope of the other line, \(h = ?\)
The circles \(x^2 + y^2 + 2x - 6y - 6 = 0\) and \(x^2 + y^2 - 6x - 2y + k = 0\) are two intersecting circles and \(k\) is not an integer. If \( \theta \) is the angle between the two circles and \( \cos \theta = -\frac{5}{24} \), then find \( k \).
If the focal chord of the parabola \( x^2 = 12y \) drawn through the point \( (3, 0) \) intersects the parabola at the points P and Q, then the sum of the reciprocals of the abscissae of the points P and Q is:
If the inverse point of the point \(P(3,3)\) with respect to the circle \(x^2 + y^2 - 4x + 4y + 4 = 0\) is \(Q(a,b)\), then \(a + 5b =\)
Length of the common chord of the circles \(x^2 + y^2 - 6x + 5 = 0\) and \(x^2 + y^2 + 4y - 5 = 0\) is:
If \( A(1, -2), B(-2, 3), C(-1, -3) \) are the vertices of a triangle ABC. \( L_1 \) is the perpendicular drawn from \( A \) to \( BC \) and \( L_2 \) is the perpendicular bisector of \( AB \). If \( (l, m) \) is the point of intersection of \( L_1 \) and \( L_2 \), then \( 26m - 3 = \)