Question

A straight line $L:4x-4y+3=0$ is rotated in clockwise about the point where the line cuts the $y$-axis and a circle $S_1$ whose centre is $\left(\lambda ,\frac{3}{4}\right)$ touches both the line $L$ and $L_1$ ( $L_1$ is the line obtained after rotation) and the $x$-axis.
If area of the triangle formed by the lines $L_1$, angle bisector between $L\&L_1$ and the $x$-axis is $\frac{p}{q},pq\in R$ then least value of $(p+q)$ equals

• A. 3
• B. 17
• C. 41
• D. 56

Answer 1

Answer: (C)

Using $p=r$
(i)$\frac{4\lambda -3+3}{4\sqrt{2}}=r=\frac{3}{4}$
$\Rightarrow \frac{\lambda }{\sqrt{2}}=\frac{3}{4}$
$\therefore \lambda =\frac{3}{2\sqrt{2}}$
$\Rightarrow [\lambda ]=1$

(ii) $\Delta OAB=\frac{1}{2}\times \frac{3}{4}\times \frac{3}{4}$
$=\frac{9}{32}\equiv \frac{p}{q}$
$\therefore (p+q)=41$