Question

The ratio of the largest and shortest distances from the point $(2, -7)$ to the circle $x^2 + y^2 - 14x - 10y - 151 = 0$ is

• A. $15 : 13$
• B. $7 : 1$
• C. $3 : 2$
• D. $14 : 1$

Hint:

Always reduce general circle form and use center-radius geometry for external point distances.

Answer 1

The Correct Option is D

Convert circle to standard form:
$x^2 + y^2 - 14x - 10y = 151$
Complete the square:
$(x - 7)^2 + (y - 5)^2 = 225 \Rightarrow$ radius = $15$, center = $(7, 5)$
Distance from $(2, -7)$ to center = $\sqrt{(7 - 2)^2 + (5 + 7)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
Largest distance = $r + d = 28$, shortest = $r - d = 2$
Ratio = $28 : 2 = 14 : 1$