JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Parabola

Let T be the tangent to the parabola ${\mathrm{y}}^2=16\mathrm{x}$ at the point (64, 32). Let L be the tangent to the same parabola at another point $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ on the parabola. If L and T are perpendicular to each other, then the distance betwe …

Let $A_1,B_1,C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1{C}_1$ and $B_1{C}_1$ are tangents to the curve $y^2=8x$ at $A_1$ and $B_1$, respectively. If $O=(0,0)$ and $C_1=(-4,0)$, then which of the following statements is (are) TRUE?

A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0,-\alpha )$ to the parabola $x^2=-4ay$, where $a>0$. Let $L$ be the line passing through $(0,-\alpha )$ and parallel to the directrix of the parabola.
Suppose that $L$ intersects the parabola at two points $A$ …

Let ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ be two distinct common tangents to the ellipse $\mathrm{E}:\frac{{\mathrm{x}}^2}{6}+\frac{{\mathrm{y}}^2}{3}=1$ and the parabola $\mathrm{P}:{\mathrm{y}}^2=12\mathrm{x}$. Suppose that the tangent ${\mathrm{T}}_1$ touches P and E at the …

Let P be a point on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$, where $\mathrm{a}>0$. The normal to the parabola at P meets the x-axis at a point Q. The area of the triangle PFQ, where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive i …

Consider the parabola $y^2=4x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P=(-2,1)$ meet the parabola at $P_1$ and $P_2$. Let $Q_1$ and $Q_2$ be points on the lines $S{P}_1$ and $S{P}_2$ respectively such that …