Straight Line and Pair of Straight Lines 1 Question 37
37. Line $L$ has intercepts $a$ and $b$ on the coordinate axes. When, the axes are rotated through a given angle, keeping the origin fixed, the same line $L$ has intercepts $p$ and $q$, then
$(1990,2 M)$
(a) $a^{2}+b^{2}=p^{2}+q^{2}$
(b) $\frac{1}{a^{2}}+\frac{1}{b^{2}}=\frac{1}{p^{2}}+\frac{1}{q^{2}}$
(c) $a^{2}+p^{2}=b^{2}+q^{2}$
(d) $\frac{1}{a^{2}}+\frac{1}{p^{2}}=\frac{1}{b^{2}}+\frac{1}{q^{2}}$
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Solution:
- Since, the origin remains the same. So, length of the perpendicular from the origin on the line in its position $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{p}+\frac{y}{q}=1$ are equal. Therefore,
$$ \frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}=\frac{1}{\sqrt{\frac{1}{p^{2}}+\frac{1}{q^{2}}}} \Rightarrow \frac{1}{a^{2}}+\frac{1}{b^{2}}=\frac{1}{p^{2}}+\frac{1}{q^{2}} $$
