SATHEE: Straight Line and Pair of Straight Lines 1 Question 37

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

(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,

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

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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

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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