List of top Mathematics Questions on sections of a cone

If the distance of a point P on an ellipse from its focus (1, 2) is half of the distance of P from its corresponding directrix \( x+y=0 \), then the point of intersection of the given directrix and its major axis, is
Let S be the focus of the parabola \( y^{2}=36x \) and let the line \( x+by+c=0 \) intersect the parabola at the points Q and R. If the centroid of \( \triangle QRS \) is (57,0), then -c is
A normal is drawn to the hyperbola \( 9x^{2}-16y^{2}=144 \) at one of the ends of its latus rectum. If that end lies in the third quadrant and the equation of the normal is \( ax+by+c=0 \) then \( \frac{b+c}{a} = \)
If a tangent drawn to the parabola \( y^{2}=16x \) meets the curve \( xy=4 \) at the points P and Q, then the locus of midpoint of PQ is
Consider the ellipses given by $x^2 + 4y^2 = 1$ and $4x^2 + y^2 = 1$.
If $\alpha$ is the area of the common region that lies inside both the given ellipses, then the value of $\tan(\alpha/2)$ is {
Consider the ellipses given by $x^2 + 4y^2 = 1$ and $4x^2 + y^2 = 1$.
Let $P$ be the point in the first quadrant where the given ellipses intersect. If $\theta$ is the acute angle between the tangents to the given ellipses at the point $P$, then the value of $4 \tan \theta$ is {
Consider the ellipse \(E\) given by \[ \frac{x^2}{18} + \frac{y^2}{12} = 1. \] Let \(H\) be the hyperbola whose eccentricity is the reciprocal of the eccentricity of \(E\) and whose foci are the same as that of \(E\). Let \(P\) and \(Q\) be the points of intersection of \(H\) and the parabola \[ \sqrt{5}\,y = x^2 \] in the first quadrant. Let \(d\) be the distance between \(P\) and \(Q\). If \(a\) and \(b\) are the integers such that \[ d^2 = a + b\sqrt{5}, \] then the value of \[ a - b \] is:
Let $T$ be the tangent to the parabola $y^2 = 16x$ at the point $(64, 32)$. Let $L$ be the tangent to the same parabola at another point $(x_1, y_1)$ on the parabola. If $L$ and $T$ are perpendicular to each other, then the distance between the point $(x_1, y_1)$ and the focus of the parabola, is
The equation of the conjugate hyperbola of the hyperbola $x^{2}-4y^{2}-2x-8y-19=0$ is
If the major axis of an ellipse subtends an angle of $120^{\circ}$ at one end of its minor axis and the length of its semi latus rectum is $\frac{4}{\sqrt{3}}$, then the sum of the lengths of its axes is
Through the focus of the parabola $x^{2}-4x-8y+44=0$, if tangents are drawn to another parabola $y^{2}=20x$, then the sum of the Y - coordinates of the points of contact of these tangents is
If $y=mx+c$, $m>0$ is a common tangent to the parabolas $y^{2}=8x$ and $y^{2}=1+4x$, then $m+c=$
The equation of the tangent to the parabola $y^2 = 8x$ which is parallel to the line $2x - y + 5 = 0$ is:
If the eccentricity of a hyperbola is $\sqrt{3}$, then the eccentricity of its conjugate hyperbola is:
The line $y = mx + c$ touches the ellipse $9x^2 + 16y^2 = 144$ if the value of $c^2$ is:
If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is:
The equation of the parabola with focus at $(3, 0)$ and directrix $x + 3 = 0$ is:
The equation of the tangent to the parabola $y^2 = 8x$ which is parallel to the line $2x - y + 5 = 0$ is:
The line $y = mx + c$ touches the ellipse $9x^2 + 16y^2 = 144$ if the value of $c^2$ is:
If the eccentricity of a hyperbola is $\sqrt{3}$, then the eccentricity of its conjugate hyperbola is:
The equation of the parabola with focus at $(3, 0)$ and directrix $x + 3 = 0$ is:
If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is:
For the parabola \( y^2 = 16x \), find the distance between the focus and the directrix.
The eccentricity of the ellipse whose major axis is three times the minor axis is:
If the eccentricity and length of latus rectum of a hyperbola are \(\frac{\sqrt{13}}{3}\) and \(\frac{10}{3}\) units respectively, then what is the length of the transverse axis?