List of top Mathematics Questions

The coordinates of the point which divides the line joining the points $ (2, 3, 4) $ and $ (3, -4, 7) $ in the ratio 2 : 4 externally is:
If the ends of the hypotenuse of a right-angled triangle are $ (0, a) $ and $ (a, 0) $, then the locus of the third vertex is:
If $ O $ is the origin and $ P, Q $ are points on the line $ 3x + 4y + 15 = 0 $ such that $ OP = OQ = 9 $, then the area of $ \triangle OPQ $ is:
Let \(Z\) be the point of intersection of the axis and the directrix of the parabola \(4x^2 - 12x + 4y + 5 = 0\). If \(S\) is its focus, then the point which divides \(SZ\) in the ratio \(2:1\) is:
The equation of the line common to the pair of lines \[ (p^2 - q^2)x^2 + (q^2 - r^2)xy + (r^2 - p^2)y^2 = 0 \] and \[ (l - m)x^2 + (m - n)xy + (n - l)y^2 = 0 \] is:
The lengths of the intercepts made by a circle \(S\) on the \(X\) and \(Y\)-axes are \(\frac{2\sqrt{13}}{3}\) and \(\frac{2\sqrt{17}}{3}\) respectively. Then the equation of the circle is:
Let the transverse axis of a hyperbola \( H \) be parallel to the X-axis and \( x^2 + y^2 - 2x - 4y + 3 = 0 \) be the equation of the auxiliary circle of \( H \). If the asymptotes of \( H \) are at right angles, then the equation of the hyperbola is:
In triangle $\triangle ABC$, $A=(1,2)$ and the equations of medians through $B$ and $C$ are $x+y=5$ and $x=4$. Find area of triangle ABC.
The line $ x + y + 2 = 0 $ intersects the circle $ x^2 + y^2 + 4x - 4y - 4 = 0 $ in two points $ A $ and $ B $. Let $ S = x^2 + y^2 + 2gx + 2fy + c = 0 $ be a different circle passing through the points $ A $ and $ B $. If the distance of the centre of $ S $ from $ AB $ is $ \sqrt{2} $, then $ g + f + c = $:
If the lines joining the origin to the points of intersection of a line $L$ and $x^2 + y^2 = 4$ are the coordinate axes, then the equation of the line $L$ is
The equation of a tangent to the circle \(x^2 + y^2 + 2x - 12y - 132 = 0\) which is perpendicular to the line \(12x + 5y + k = 0\) is:
If the line passing through the points \( (5,1,a) \) and \( (3,b,1) \) crosses the YZ plane at the point \( \left( 0, \frac{17}{2}, \frac{-13}{2} \right) \), then \( a + b = \):
Let the locus of the point of intersection of the perpendicular tangents drawn to the circle \[ x^2 + y^2 + 6x - 4y - 12 = 0 \] be the circle \( S \). Then the equation of the tangent drawn to \( S \) which is perpendicular to the line \( 6x - 4y + k = 0 \) is:
If the area of the triangle formed by the lines \( y = x + c \) and \( 2x^2 + 5xy + 3y^2 = 0 \) is \( \frac{1}{20} \) sq. units, then \( c = \)
If $x+y-1=0$ and $2x - y + 1=0$ are conjugate lines with respect to the circle $x^2 + y^2 - 4x + 2fy - 1 = 0$, then $f=$
If \(2x^2 - 5xy + 2y^2 = 0\) represents two sides of a triangle whose centroid is \((1,1)\), then the equation of the third side is:
Let the curve \(x^2 + 2y^2 = 2\) intersect the line \(x + y = 1\) at two points \(P\) and \(Q\) and \(O\) be the origin. If \(\theta\) is the acute angle between the lines \(OP\) and \(OQ\), then \(\tan \theta =\)
If \( (a^2 - 1)x + ay + (3 - a) = 0 \) is a normal to the curve \( xy = 1 \), then the interval in which ‘a’ lies is:
The product of the slopes of the common tangents drawn to the circles $x^2+y^2+2x-2y-2=0$ and $x^2+y^2-2x+2y+1=0$ is
If the chord through $(1,-2)$ cuts the curve $3x^2 - y^2 - 2x + 4y = 0$ at P and Q, then the angle subtended by PQ at the origin is
The line $3x+y-5=0$ touches a circle S at $(1,2)$. If $(h,k)$ is the ntre of the circle S such that $h^2+hk+k^2=37$ and radius is $\sqrt{10}$, then $k=$
Two sides of a rhombus are along the lines \( x - y + 1 = 0 \) and \( 7x - y - 5 = 0 \). If its diagonals intersect at \( (-1, -2) \), then one of the vertices of this rhombus is:
In \( \triangle ABC \), the coordinates of the vertices are \( A(1,8,4) \), \( B(0,-11,4) \), and \( C(2,-3,1) \). If \( D \) is the foot of the perpendicular drawn from \( A \) to \( BC \), then the coordinates of \( D \) are:
Let X-axis be the transverse axis and Y-axis be the conjugate axis of a hyperbola H. Let \( x^2 + y^2 = 16 \) be the director circle of H. If the perpendicular distance from the centre of H to its latus rectum is \( \sqrt{34} \), then \( a + b = \):
Tangent \( L_1 = 3x - 4y - 8 = 0 \) and the chord \( L_2 = x + y - 1 = 0 \) are at a distance of 2 and \( \sqrt{2} \) units respectively from the centre of a circle \( S \). \((h, k)\) is the centre of \( S \) such that \( h^2 + k^2 = 13 \). If the midpoint of the chord \( L_2 = 0 \) is \( (\alpha, \beta) \) and the radius of the circle is \( r \), then \( \alpha + \beta + r = \):