Question

If p be the measure of perpendicular segment from the origin on the line whose intercepts on the axes are 'a' and 'b', then which of the following is/are correct?

• A. $\frac{1}{a^2}=\frac{1}{p^2}+\frac{1}{b^2}$
• B. $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$
• C. $p^2(a^2+b^2)=a^2{b}^2$
• D. $p^2\left(a^2{b}^2\right)=a^2+b^2$

Answer 1

The correct option is B $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$

Equation of line in perpendicular form is:

$x\cos \alpha +y\sin \alpha =p$

where p$=$length of perpendicular segment

$\alpha =$angle made by perpendicular with axis.

at $y=0$

$\Rightarrow x=\frac{p}{\cos \alpha }=a$

$\Rightarrow \frac{p}{a}=\cos \alpha$ ………..$\left(1\right)$

at $x=0$

$\Rightarrow y=\frac{p}{\sin \alpha }=b$

$\Rightarrow \frac{p}{b}=\sin \alpha$ ……..$\left(2\right)$

Now, ${\sin }^2\alpha +{\cos }^2\alpha =1$

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

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