First slide

Equation of line in Straight lines

Question

A line L has intercepts a and b on the coordinate axes. When the axes are rotated through an angle, keeping the origin fixed, the same line L has intercepts p and q. Then,

Moderate

A

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

$\frac{1}{a^{2}} - \frac{1}{b^{2}} = \frac{1}{p^{2}} - \frac{1}{q^{2}}$
C

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

none of these

Solution

Since the line L has intercepts a and b on the coordinate
axes, therefore its equation is
$\frac{x}{a} + \frac{y}{b} = 1$ (1)
When the axes are rotated, its equation with respect to
the new axes and same origin will become
$\frac{x}{p} + \frac{y}{q} = 1$ (2)
In both the cases, the length of the perpendicular from the origin to the line will be same.
$∴ \frac{1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}}}} = \frac{1}{\sqrt{\frac{1}{p^{2}} + \frac{1}{q^{2}}}}$
$o r \frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{p^{2}} + \frac{1}{q^{2}}$
which is the required relation.

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