Question

If $p$ is the length of perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$ respectively, then $\frac{1}{a^2} + \frac{1}{b^2}$ equals

• A. $p^{2}$
• B. $\frac{2}{p^{2}}$
• C. $\frac{1}{p^{2}}$
• D. $\frac{1}{2p^{2}}$

Hint:

Geometry Tip: This is a standard geometric property connecting intercepts to origin distance. Memorizing $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}$ can instantly solve similar optimization or locus problems.

Answer 1

The Correct Option is C

Concept: Coordinate Geometry - Straight Lines (Intercept Form and Perpendicular Distance Formula).

Step 1: Write the equation of the straight line. The equation of a line making $x$-intercept '$a$' and $y$-intercept '$b$' is given by the intercept form: $\frac{x}{a} + \frac{y}{b} = 1$. Let's rewrite this in the standard general form $Ax + By + C = 0$ as: $\left(\frac{1}{a}\right)x + \left(\frac{1}{b}\right)y - 1 = 0$.

Step 2: Recall the perpendicular distance formula. The perpendicular distance '$d$' from the origin $(0, 0)$ to any line $Ax + By + C = 0$ is found using the formula $d = \frac{|A(0) + B(0) + C|}{\sqrt{A^2 + B^2}} = \frac{|C|}{\sqrt{A^2 + B^2}}$.

Step 3: Apply the formula to our specific line. We are given that the perpendicular distance from the origin is '$p$'. Substituting our line's coefficients $A = \frac{1}{a}$, $B = \frac{1}{b}$, and $C = -1$ into the distance formula gives: $p = \frac{|-1|}{\sqrt{\left(\frac{1}{a}\right)^2 + \left(\frac{1}{b}\right)^2}}$.

Step 4: Simplify the equation by squaring both sides. First, simplify the numerator and the terms inside the square root: $p = \frac{1}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2}}}$. To eliminate the square root, square both sides of the equation: $p^2 = \frac{1}{\frac{1}{a^2} + \frac{1}{b^2}}$.

Step 5: Rearrange to solve for the required expression. Multiply both sides by $\left(\frac{1}{a^2} + \frac{1}{b^2}\right)$ to get it out of the denominator: $p^2 \left(\frac{1}{a^2} + \frac{1}{b^2}\right) = 1$. Finally, divide both sides by $p^2$ to isolate the target expression: $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{p^2}$. $$ \therefore \text{The value of } \frac{1}{a^{2}}+\frac{1}{b^{2}} \text{ is } \frac{1}{p^{2}}. $$