Dr. SK Goyal Solutions for Chapter: Parabola, Exercise 3: INTRODUCTORY EXERCISE 5.3

If $m_1,m_2$ are slopes of the two tangents that are drawn from $\left(2,3\right)$ to the parabola $y^2=4x$ then the value of $\frac{1}{m_1}+\frac{1}{m_2}$ is:

If $\left(a, b\right)$ is the mid-point of chord passing through the vertex of the parabola $y^2=4x$, then :

The diameter of the parabola $y^{2}=6 x$ corresponding to the system of parallel chords $3x-y+c=0$ is :

For parabola $x^2+y^2+2xy-6x-2y+3=0$, the focus is :

A ray of light moving parallel to the $x$-axis gets reflected from a parabolic mirror whose equation is ${\left(y-2\right)}^2=4\left(x+1\right)$. After reflection, the ray must pass through the point:

Prove that the locus of the point of intersection of tangents to the parabola $y^2=4ax$ which meet at an angle $\alpha$ is

${\left(x+a\right)}^2{\tan }^2\alpha =y^2-4ax$.

Show that the polar of any point on the circle $x^2+y^2-2ax-3a^2=0$ with respect to the circle $x^2+y^2+2ax-3a^2=0$ will touch the parabola $y^2+4ax=0$

From the point $P\left(-1,2\right)$ tangents are drawn to the parabola $y^2=4x$. Find the equation of the chord of contact. Also, find the area of the triangle formed by the chord of contact and the tangents.