Question

Let $C$ be the locus of the mirror image of a point on the parabola $y^2=4x$ with respect to the line$y=x$. Then the equation of tangent to $C$at $P\left(2,1\right)$ is:

• A. $2x+y=5$
• B. $x+2y=4$
• C. $x+3y=5$
• D. $x-y=1$

Answer 1

The correct option is D

$x-y=1$

Explanation for the correct option:

Finding the equation of the tangent at the given point:

Consider the given equation$y^2=4x$

Its mirror image of the parabola is $x^2=4y$

Differentiating the above equation with respect to $x$

$$\begin{gathered} 2x=4\frac{dy}{dx} \\ \frac{dy}{dx}=\frac{x}{2} \end{gathered}$$

Then the equation of the tangent to curve at $P\left(2,1\right)$

$$\begin{gathered} {\overline{)\frac{dy}{dx}}}_{P(2,1)}=\frac{2}{2} \\ {\overline{)\frac{dy}{dx}}}_{P(2,1)}=1\dots \dots \left(i\right) \end{gathered}$$

We know that the equation of tangent is,

$$\begin{gathered} y-y_1=m\left(x-x_1\right) \\ y-1=1\left(x-2\right)[y_1=1,x_1=2,m=1\text{from equation}\left(i\right)] \\ y-1=x-2 \\ x-y=2-1 \\ x-y=1 \end{gathered}$$

Hence , the correct answer is Option (D).