Circle 1 Question 13
13. 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 $x^{2}+y^{2}-x+3 y=0$ 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) $x+7 y=0$
(d) $x-7 y=0$
$(1999,1 M)$
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Answer:
Correct Answer: 13. (a, c)
Solution:
- Let equation of line $L _1$ be $y=m x$. Intercepts made by $L _1$ and $L _2$ on the circle will be equal i.e. $L _1$ and $L _2$ are at the same distance from the centre of the circle;
Centre of the given circle is $(1 / 2,-3 / 2)$. Therefore,
$$ \begin{aligned} & & \frac{|1 / 2-3 / 2-1|}{\sqrt{1+1}} & =\left|\frac{\frac{m}{2}+\frac{3}{2}}{\sqrt{m^{2}-1}}\right| \Rightarrow \frac{2}{\sqrt{2}}=\frac{|m+3|}{2 \sqrt{m^{2}+1}} \\ \Rightarrow & & 8 m^{2}+8 & =m^{2}+6 m+9 \\ \Rightarrow & & 7 m^{2}-6 m-1 & =0 \Rightarrow(7 m+1)(m-1)=0 \\ & & m & =-\frac{1}{7}, m=1 \end{aligned} $$
Thus, two chords are $x+7 y=0$
$$ \text { and } \quad x-y=0 \text {. } $$
Therefore, (b) and (c) are correct answers.
