Dr. SK Goyal Solutions for Chapter: Parabola, Exercise 5: EXERCISE ON LEVEL-II

Find the equation of circle, which touches the $y$-axis and the parabolas $y^2=x$ and $2y=2x^2-5x+1$ only at the point when they touch each other and at no other point.

Find the locus of the point of intersection of tangents drawn at the end points of a variable chord of the parabola $y^2=4ax$, which subtends a constant angle ${\tan }^{-1}\left(b\right)$ at the vertex of the parabola.

The sides of a triangle touch $y^2=4ax$ and two of its angular points lie on $y^2=4b\left(x+c\right)$. Show that the locus of the third angular point "
$a^2{y}^2=4{\left(2b-a\right)}^2\left(ax+4bc\right)$.

$PC$ is the normal at $P$ to the parabola $y^2=4ax,C$ being on the axis. $CP$ is produced outwards to $Q$ so that $PQ=CP$; Show that the locus of $Q$ is a parabola and that the locus of the intersection of the tangents at $P$ and $Q$ to the parabola on which they lie is
$y^2\left(x+4a\right)+16a^3=0$.

Let $y=f\left(x\right)$ be a curve. Lines are drawn through any point $\left(x,y\right)$ on the curve parallel to the axes of reference the rectangle formed by the axes of reference and the lines drawn in two parts, the ratio of whose areas is $2:1$, then prove that the curve must be a parabola.

Prove that the foot of any perpendicular from the point $\left(0,-c\right),c>0$ to any normal to the parabola $x^2=4ay$ lies on the curve whose equation is
$x^4=\left(y+c\right)\left\{x^2\left(2a-y\right)+a{\left(y+c\right)}^2\right\}$.

Prove that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other.

Find the vertex, the length of the latus rectum and the equations of the axis and tangent at the vertex of the parabola
${\left(x-y+2\right)}^2=8\sqrt{2}\left(x+y-6\right)$