Question

A line makes intercepts h and K on the coordinate axis. If p is the length of the perpendicular from the origin to the line then show that 1/h²+1/k²=1/p²

Answer 1

A line makes intercepts h and K on the coordinate axis. If p is the length of the perpendicular from the origin to the line then show that 1/h²+1/k²=1/p²

The proof is given below:

If the length of perpendicular on  a line is p and its inclination from the positive direction of x-axis is is $\theta$ then the equation of the line is given by

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

Writing this equation in slope intercept form

$\frac{x}{\sec\theta}+\frac{y}{\cosec\theta}=p$

or, $\frac{x}{p\sec\tehta}+\frac{y}{p\cosec\theta}=1$

But given that x-intercept is h and y-intercept is k

Therefore,

$h=p\sec\theta$

$\implies \frac{1}{h}=\frac{1}{p\sec\theta}$

$\implies \frac{1}{h}=\frac{1}{p}\cos\tehta$

Similarly, we can show that

$\frac{1}{k}=\frac{1}{p}\sin\theta$

But we know that

$\sin^2\theta+\cos^2\theta=1$

Thus,

$(\frac{1}{p})^2(\sin^2\theta+\cos^2\theta)=(\frac{1}{k})^2+(\frac{1}{k})^2$

$\implies \frac{1}{h^2}+\frac{1}{k^2}=\frac{1}{p^2}$    (Hence Proved)

Hope this helps.

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Answer 2

Hence proved,

1/h² + 1/k² = 1/p²

Hope It Will Help You..