Question

For any two nonzero real numbers $a$ and $b$, if the line $\dfrac{x}{a} + \dfrac{y}{b} = 1$ is a tangent to the circle $x^2 + y^2 = 1$, then which of the following is true?

• A. $\left(\dfrac{1}{a}, \dfrac{1}{b}\right)$ lies inside the circle
• B. $(a, b)$ lies inside the circle
• C. $\left(\dfrac{1}{a}, \dfrac{1}{b}\right)$ lies on the circle
• D. $(a, b)$ lies on the circle

Hint:

To check tangency, convert line to normal form and compare perpendicular distance from circle center to radius.

Answer 1

The Correct Option is C

The line $\dfrac{x}{a} + \dfrac{y}{b} = 1$ has intercepts $a$ and $b$.
A line is tangent to the circle $x^2 + y^2 = 1$ if the perpendicular distance from the origin is 1.
This perpendicular distance = $\dfrac{1}{\sqrt{\left(\dfrac{1}{a}\right)^2 + \left(\dfrac{1}{b}\right)^2}}$
So, $\left(\dfrac{1}{a}\right)^2 + \left(\dfrac{1}{b}\right)^2 = 1$
$\Rightarrow \left(\dfrac{1}{a}, \dfrac{1}{b}\right)$ lies on the circle $x^2 + y^2 = 1$