Embibe Experts Solutions for Chapter: Parabola, Exercise 3: Level 3

The tangent and normal at the point $P\left(a{t}^2,2at\right)$ to the parabola $y^2=4ax$ meet the $x$-axis in $T$ and $G\mathit{,}$ respectively, then the angle at which the tangent at $P$ to the parabola is inclined to the tangent at $P$ to the circle through $P\mathit{,}\mathit{ }T\mathit{,}\mathit{ }G$ is

Through the vertex $O$ of the parabola $y^2=4ax,$ two chords $OP$ and $OQ$ are drawn and the circles on $OP$ and $OQ$ as diameters intersect in $R.$ If ${\theta }_1, {\theta }_2$ and $\varphi$ are the angles made with axis by the tangents at $P$ and $Q$ on the parabola and by $OR,$ then the value of $\cot {\theta }_1+\cot {\theta }_2$

$AB$ is a double ordinate of the parabola $y^2=4ax.$ Tangents drawn to parabola at $A$ and $B$ meet $y$-axis at $A_1$ and $B_1$, respectively. If the area of trapezium $A{A}_1{B}_{1}B$ is equal to $24a^2,$ then angle subtended by $A_1{B}_1$ at the focus of the parabola is equal to

Tangent and normal are drawn at $P\left(16,16\right)$ on the parabola $y^2=16x$, which intersect the axis of the parabola at $A$ and $B$, respectively. If $C$ is the centre of the circle through the points $P\mathit{,}\mathit{ }A$ and $B$ and $\angle CPB=\theta$, then the value of $\tan \theta$ is

Length of the shortest normal chord of the parabola $y^2=4ax$ is

The locus of the centre of the circle $C$, which intersect orthogonally and also passing through the focus of the parabola $y^2=4x$ is 

All the three vertices of an equilateral triangle lie on the parabola $y=x^2$ and one of its sides has a slope of $2.$ Then the sum of the $x$-coordinates of the three vertices is

If the normals to the parabola $y^2=4ax$ at three points $P, Q$ and $R$ meet at $A$ and $S$ be the focus, then $SP·SQ·SR$ is equal to