Question

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 has intercepts c and d, then-

• A. a² + b² = c² + d²
• B. $\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{c^{2}} + \frac{1}{d^{2}}$
• C. a² + c² = b² + d²
• D. $\frac{1}{a^{2}} + \frac{1}{c^{2}} = \frac{1}{b^{2}} + \frac{1}{d^{2}}$

Answer 1

Answer: (B)

$\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{c^{2}} + \frac{1}{d^{2}}$

Answer 2

Equation of the line in the two frames are

$\frac{x}{a} + \frac{y}{b}$– 1 = 0 and $\frac{x'}{c} + \frac{y'}{d}$– 1 = 0

The distance of the origin from the line is same in both the frames, therefore we have

$\frac{1}{\left( \frac{1}{a} \right)^{2} + \left( \frac{1}{b} \right)^{2}} = \frac{1}{\left( \frac{1}{c} \right)^{2} + \left( \frac{1}{d} \right)^{2}}$

i.e. $\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{c^{2}} + \frac{1}{d^{2}}$.