If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x² + y² = 1 is a circle of radius r, then r is equal to :

Solution

Maths sir · 3 steps

Name the midpoint (h, k), write the circle point in terms of h and k, and push it back into the circle.

1. Set up the midpoint. Let (h, k) be the midpoint of the segment joining (3, 2) to a point (x₁, y₁) on the circle. Then h = (x₁ + 3)/2 and k = (y₁ + 2)/2.
2. Find the circle point. Solving gives x₁ = 2h - 3 and y₁ = 2k - 2. This point lies on x² + y² = 1, so it must satisfy that equation.

   (2h-3)² + (2k-2)² = 1 ⇒ (h-3/2)² + (k-1)² = 1/4

3. So. The locus is a circle with centre (3/2, 1) and r² = 1/4, so r = 1/2. The answer is (b).

The idea to keep. A midpoint shrinks everything by half: the locus of midpoints from a fixed point to a circle of radius a is a circle of radius a/2, centred halfway to the old centre.

Answer from the NTA final answer key. Solution by PrepWiser.