Let the mirror image of a circle c₁: x²+y²-2 x-6 y+α=0 in line y=x+1 be c₂: 5 x²+5 y²+10 g x+10 f y+38=0. If r is the radius of circle c₂, then α+6 r² is equal to ________.

Solution

Maths sir · 5 steps

Reflection moves the centre but keeps the radius, so reflect the centre and match the two radii.

1. Read off c1. For x²+y²-2x-6y+α=0 the centre is (1, 3) and r² = 1 + 9 - α = 10 - α.
2. Reflect the centre. Write the line as x - y + 1 = 0. At (1, 3) this expression is 1 - 3 + 1 = -1. Use the image formula below.

   (x-1)/1 = (y-3)/-1 = -2(-1)/(1²+(-1)²) = 1 ⇒ (x, y) = (2, 2)

3. Read off c2. Divide c₂ by 5 to get x²+y²+2gx+2fy+38/5=0. Its centre (-g, -f) must be (2, 2), so g = f = -2. Then r² = 4 + 4 - 38/5 = 2/5.
4. Match the radii. A mirror image has the same radius, so 10 - α = 2/5. This gives α = 48/5.
5. So. α + 6r² = 48/5 + 12/5 = 60/5 = 12.

The common slip: 22. Reflecting the centre in y = x instead of y = x + 1 gives (3, 1). That leads to r² = 12/5 and α = 38/5, so the total comes out as 22.

The idea to keep. Reflecting a circle in a line only moves the centre to its mirror point. The radius stays the same. Make x² and y² have coefficient 1 before you read off the centre and radius.

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