JEE MainMathematicsModerateStraight Lines

The line L has intercepts $a$ and $b$ on the coordinate axes. The
coordinate axes are rotated through a fixed angle, keeping the origin
fixed. If $p$ and $q$ are the intercepts of the line $L$ on the new
axes, then
$\frac{1}{a^{2}} - \frac{1}{p^{2}} + \frac{1}{b^{2}} - \frac{1}{q^{2}}$
is equal to

Show answer and solution

Equation of the line

L

in the two coordinate systems is

\frac{x}{a} + \frac{y}{b} = 1,\frac{X}{p} + \frac{Y}{q} = 1

where (X,
Y) are the new coordinates of a point

(x,\ \ y)

when the axes are
rotated through a fixed angle, keeping the origin fixed. as the length
of the perpendicular from the origin has not changed.

\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}}

\Rightarrow \frac{1}{a^{2}} - \frac{1}{p^{2}} + \frac{1}{b^{2}} - \frac{1}{q^{2}} = 0

Source: Grass-G / M-19 Mathematics Paper 28 Nov Fourth.docx

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