Question

The least distance of the point A(10, 7) from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ is length of seg AM. If MM' is the diameter of the circle, then the lengths of AM and AM' are respectively ______, ______ units.

• A. 5, 10
• B. 5, 15
• C. 4, 15
• D. 2, 10

Hint:

From an external point to a circle:
Minimum Distance = (Distance to Center) - Radius.
Maximum Distance = (Distance to Center) + Radius.
These two extreme points always lie perfectly collinear with the center of the circle!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a point A located externally to a circle. The line passing through A and the center of the circle intersects the circle at two points: M (the nearest point) and M' (the furthest point). We must calculate the lengths of the segments AM and AM'.

Step 2: Detailed Explanation:
First, identify the center and radius of the given circle.
The general equation is $x^2 + y^2 + 2gx + 2fy + c = 0$.
Comparing to $x^2 + y^2 - 4x - 2y - 20 = 0$:
$2g = -4 \implies g = -2$
$2f = -2 \implies f = -1$
$c = -20$
The center of the circle $C$ is at $(-g, -f) = (2, 1)$.
The radius $R$ is $\sqrt{g^2 + f^2 - c} = \sqrt{(-2)^2 + (-1)^2 - (-20)} = \sqrt{4 + 1 + 20} = \sqrt{25} = 5$.
Now, calculate the distance from point $A(10, 7)$ to the center $C(2, 1)$ using the distance formula:
$AC = \sqrt{(10 - 2)^2 + (7 - 1)^2}$
$AC = \sqrt{8^2 + 6^2}$
$AC = \sqrt{64 + 36} = \sqrt{100} = 10$.
The line segment originating from A and passing straight through the center C intersects the circle edge first at point M, and then emerges out the far side at point M'.
Therefore, the points line up in the order A $\rightarrow$ M $\rightarrow$ C $\rightarrow$ M'.
- The distance to the nearest point M (Least distance) is the total distance to the center minus the radius:
$AM = AC - R = 10 - 5 = 5$.
- The distance to the furthest point M' (Greatest distance) is the total distance to the center plus the radius:
$AM' = AC + R = 10 + 5 = 15$.
The lengths are 5 and 15 units respectively.

Step 3: Final Answer:
The lengths of AM and AM' are 5, 15, matching option (b).