Question

The intercept on x-axis made by tangent to the curve, $y={\int }_0^x\left|t\right|dt,x\in R$, which are parallel to the line $y=2x$, are equal to

• A. $\pm 1$
• B. $\pm 2$
• C. $\pm 3$
• D. $\pm 4$

Answer 1

The correct option is A $\pm 1$

$y={\int }_0^x\left|t\right|dt$

$\Rightarrow \frac{dy}{dx}=\left|x\right|$

Since tangent to the curve is parallel to the line $y=2x$.

$\therefore \frac{dy}{dx}=2$

$\Rightarrow x=\pm 2$

Therefore,

$y={\int }_0^{\pm 2}\left|t\right|dt$

$\Rightarrow y=\pm 2$

Therefore,

Equation of tangents are-

$y-2=2(x-2)$

$y+2=2(x+2)$

For $x$-intercept, substitute $y=0$, we have

$0-2=2(x-2)\Rightarrow x=+1$

$0+2=2(x+2)\Rightarrow x=-1$

Hence the intercept on $x$-axis are $\pm 1$.