Question

Let $L_1$ be a straight line passing through the origin and $L_2$ be the straight line $x+y=1$. If the intercepts made by the circle $x^2+y^2-x+3y=0$ on $L_1$ and $L_2$ are equal, then which of the following equation can represent $L_1$?

• A. $x+y=0$
• B. $x-y=0$
• C. $7y+2x=0$
• D. $x-7y=0$

Answer 1

Answer: (B)

$x-y=0$

Let the equation of line $L_1$ be $y=mx.$

Intercpts made by $L_1$ and $L_2$ on the circle will be equal if $L_1$ and $L_2$ are at the same distance from the centre of the circle .

Centre of the given circle is $(\frac{1}{2},\frac{-3}{2}).$

Therefore, $\frac{|\frac{1}{2}-\frac{3}{2}-1|}{\sqrt{1+1}}=\frac{|\frac{m}{2}+\frac{3}{2}|}{\sqrt{1+m^2}}\Rightarrow \frac{4}{\sqrt{2}}=\frac{m+3}{\sqrt{1+m^2}}$

$\Rightarrow 7m^2-6m-1=0$

$\Rightarrow (7m+1)(m-1)=0$

$\Rightarrow m=1,\frac{-1}{7}.$

Thus, two chords are $y=x$ and $7y+x=0.$