Question

If the curve $y=|x-3|$ touches the parabola $y^2=\lambda (x-4),\lambda >0,$ then length of latus rectum of the parabola(in units) is

• A. $2$
• B. $4$
• C. $16$
• D. $8$

Answer 1

The correct option is B $4$

The curve $y=|x-3|$ can be rewritten as
$y=\{\begin{array}{c}x-3,x\ge 3\\ 3-x,x<3\end{array}$
Hence curve will touch the parabola $y^2=\lambda (x-4)$ for $x\ge 3$ as $\lambda >0$

Now, let $X=x-4$, then
$y^2=\lambda X$ and $y=X+1$

Now, using the tangency condition
$m=1,c=1\Rightarrow c=\frac{a}{m}\Rightarrow \frac{\lambda }{4}=1\Rightarrow \lambda =4$

Hence, the length of latus rectum is $4\text{units}$