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:

  1. 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,

$$ \frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}=\frac{1}{\sqrt{\frac{1}{p^{2}}+\frac{1}{q^{2}}}} \Rightarrow \frac{1}{a^{2}}+\frac{1}{b^{2}}=\frac{1}{p^{2}}+\frac{1}{q^{2}} $$



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