Question

Find the least distance from the point $ (10, 7) $ to the circle: $$ x^2 + y^2 - 4x - 2y - 20 = 0 $$

• A. 6
• B. 7
• C. 4
• D. 5

Hint:

To find the shortest distance from a point to a circle, subtract the radius from the distance between the point and center.

Answer 1

The Correct Option is D

We first convert the given circle equation into standard form by completing the square: \[ x^2 - 4x + y^2 - 2y = 20 \Rightarrow (x - 2)^2 - 4 + (y - 1)^2 - 1 = 20 \Rightarrow (x - 2)^2 + (y - 1)^2 = 25 \] So, the circle has: - Center: \( C = (2, 1) \) - Radius: \( r = \sqrt{25} = 5 \) Now, find distance from point \( P = (10, 7) \) to the center \( C \): \[ PC = \sqrt{(10 - 2)^2 + (7 - 1)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \] Least distance from point to circle = \( PC - r = 10 - 5 = \boxed{5} \)