If the curves, x² – 6x + y² + 8 = 0 and
x² – 8y + y² + 16 – k = 0, (k > 0) touch each other at a point, then the largest value of k is ______.

Solution

Maths sir · 4 steps

Touching circles can touch from outside or from inside, so check both cases and take the bigger k.

1. Complete the squares. The first curve is (x-3)² + y² = 1, with centre C₁(3, 0) and radius r₁ = 1. The second is x² + (y-4)² = k, with centre C₂(0, 4) and radius r₂ = √k.
2. Distance between centres. C₁C₂ = √(3² + 4²) = 5. Two circles touch when this distance equals the sum of the radii (external touch) or the difference of the radii (internal touch).

   √k + 1 = 5 or |√k - 1| = 5

3. Solve both cases. External touch gives √k = 4, so k = 16. Internal touch gives √k = 6, so k = 36; here the small circle sits inside the big one and touches it from inside. The other root, √k = -4, is rejected because a radius cannot be negative.
4. So. The possible values are 16 and 36, and the largest value of k is 36.

The common slip: 16. Using only external touching, √k + 1 = 5, gives 16. Internal touching gives the larger value, 36.

The idea to keep. For touching circles, always write both conditions: C₁C₂ = r₁ + r₂ and C₁C₂ = |r₁ - r₂|. A question asking for the largest or smallest value is hinting that both cases apply.

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