Consider a circle C₁: x²+y²-4 x-2 y=α-5. Let its mirror image in the line y=2 x+1 be another circle C₂: 5 x²+5 y²-10 f x-10 g y+36=0. Let r be the radius of C₂. Then α+r is equal to _________.

Solution

Maths sir · 4 steps

Reflection moves the centre but keeps the radius, so find the image centre and compare radii.

1. Read off C1. Writing C₁ as (x-2)² + (y-1)² = α gives centre (2, 1) and radius² = α.
2. Reflect the centre. Write the line as 2x - y + 1 = 0. At (2,1) the left side is 4 - 1 + 1 = 4, so the image (h, k) satisfies the formula below.

   (h-2)/2 = (k-1)/-1 = -2(4)/(2²+1²) = -8/5 ⇒ (h,k) = (-6/5, 13/5)

3. Radius of C2. Divide C₂ by 5 to get x² + y² - 2fx - 2gy + 36/5 = 0. Its centre is (f, g) = (-6/5, 13/5). So r² = 36/25 + 169/25 - 180/25 = 1, which gives r = 1.
4. So. A mirror image has the same radius, so α = r² = 1. That gives α + r = 1 + 1 = 2.

The idea to keep. A circle's mirror image has the same radius and a reflected centre. Always make the coefficient of x² equal to 1 before you read the centre and radius.

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