Question

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

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

Hint:

This problem represents a classic, standard coordinate geometry identity.
Think of it as a 2D right-angled triangle analogy: the area of a right-angled triangle with legs $a$ and $b$ and altitude to hypotenuse $p$ satisfies the inverse Pythagorean relation: $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}$. Memorizing this form saves considerable derivation time!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a line with axes intercepts $a$ and $b$. The shortest geometric distance from the coordinate origin $(0,0)$ to this line is defined as $p$. We need to identify the correct mathematical expression for $\frac{1}{a^2} + \frac{1}{b^2}$.

Step 2: Key Formula or Approach:
The equation of a straight line expressed in its standard intercept form is:
$$\frac{x}{a} + \frac{y}{b} = 1 \implies \frac{1}{a}x + \frac{1}{b}y - 1 = 0$$ The perpendicular distance formula from an external point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is:
$$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$ We evaluate this distance formula from the origin point $(0,0)$.

Step 3: Detailed Explanation:
Substitute the coordinates of the origin $(0,0)$ into our perpendicular distance equation where $A = \frac{1}{a}$, $B = \frac{1}{b}$, and $C = -1$:
$$p = \frac{\left|\frac{1}{a}(0) + \frac{1}{b}(0) - 1\right|}{\sqrt{\left(\frac{1}{a}\right)^2 + \left(\frac{1}{b}\right)^2}}$$ $$p = \frac{|-1|}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2}}} = \frac{1}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2}}}$$ To isolate the required expression, square both sides of the equation:
$$p^2 = \frac{1}{\frac{1}{a^2} + \frac{1}{b^2}}$$ Taking the reciprocal of both sides yields the final equation:
$$\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{p^2}$$

Step 4: Final Answer:
The value of the expression is equal to $\frac{1}{p^2}$, which corresponds precisely to option (D).