If $p$ is the length of perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$, then show that $\dfrac{1}{p^2} = \dfrac{1}{a^2} + \dfrac{1}{b^2}$.

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Accepted Answer

> [!TLDR]
> The perpendicular distance from the origin to a line with intercepts $a$ and $b$ satisfies $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}$ because the intercept form of the line and the perpendicular distance formula combine to yield this elegant relationship.

The intercept form of a line is one of the most natural ways to describe a line when you know where it crosses the axes. A line with $x$-intercept $a$ and $y$-intercept $b$ can be written as

$$\frac{x}{a} + \frac{y}{b} = 1$$

This equation tells us that when $x = a$, we have $y = 0$ (the $x$-intercept), and when $y = b$, we have $x = 0$ (the $y$-intercept). The perpendicular distance from a point to a line is the shortest distance, and there's a standard formula for it. Our task is to connect these two ideas.

## Finding the perpendicular distance

1. **Convert the intercept form to standard form**

Starting with $\frac{x}{a} + \frac{y}{b} = 1$, multiply through by $ab$ to clear denominators:
   
   $$bx + ay = ab$$
   
   Rearranging to the form $Ax + By + C = 0$:
   
   $$bx + ay - ab = 0$$
   
   Here $A = b$, $B = a$, and $C = -ab$.

2. **Apply the perpendicular distance formula**

The perpendicular distance from a point $(x_0, y_0)$ to the line $Ax + By + C = 0$ is given by
   
   $$p = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$
   
   For the origin $(0, 0)$:
   
   $$p = \frac{|b(0) + a(0) - ab|}{\sqrt{b^2 + a^2}} = \frac{|-ab|}{\sqrt{a^2 + b^2}} = \frac{|ab|}{\sqrt{a^2 + b^2}}$$

> [!FORMULA]
> $$p = \frac{|ab|}{\sqrt{a^2 + b^2}}$$

3. **Square both sides to eliminate the square root**

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

4. **Take the reciprocal of both sides** …

Q.If $p$ is the length of perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$ , then show that $\dfrac{1}{p^2} = \dfrac{1}{a^2} + \dfrac{1}{b^2}$ .

Concept understanding — Distance From Point To Line

Distance from a Point to a Line

The idea

Example

Finding the perpendicular distance

Q.If ppp is the length of perpendicular from the origin to the line whose intercepts on the axes are aaa and bbb, then show that 1p2=1a2+1b2\dfrac{1}{p^2} = \dfrac{1}{a^2} + \dfrac{1}{b^2}p21​=a21​+b21​.

Finding the perpendicular distance

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Q.If $p$ is the length of perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$ , then show that $\dfrac{1}{p^2} = \dfrac{1}{a^2} + \dfrac{1}{b^2}$ .

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