Question

A tangent to the curve $y^2=4x$ which is parallel to the line $2x-y=3$ has $y$ intercept equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{7}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{2}$

$y^2=4x$ ---(i)

differentiating w.r.t.$x$

$\Rightarrow 2y\frac{dy}{dx}=4$

$\Rightarrow \frac{dy}{dx}=\frac{2}{y}$

Now, $2x-y=3\Rightarrow y=2x-3$

Slope of this line is 2

Since tangent is parallel to this line

So, $\frac{dy}{dx}=2$

$\Rightarrow \frac{2}{y}=2$

$\Rightarrow y=1$

putting $y=1$ in (i)

$1=4x\Rightarrow x=\frac{1}{4}$

Therefore, tangent is at point $\left(\frac{1}{4},1\right)$

Equation of tangent at point point $\left(\frac{1}{4},1\right)$ with slope $2$ is

$y-1=2\left(x-\frac{1}{4}\right)$

$\Rightarrow y-1=2x-\frac{1}{2}$

$\Rightarrow y=2x+\frac{1}{2}$

On comparing it with $y=mx+c$ we get $c=\frac{1}{2}$

Hence $y$ intercept of the tangent is $\frac{1}{2}$