Let y = mx + c, m > 0 be the focal chord of y² = - 64x, which is tangent to (x + 10)² + y² = 4. Then, the value of 4√2 (m + c) is equal to _____________.

Solution

Maths sir · 5 steps

Put the focus on the line to tie c to m, then set its distance from the circle's centre to 2.

1. Find the focus. Compare y² = -64x with y² = -4ax. This gives 4a = 64, so a = 16, and the focus is (-16, 0).
2. Focal chord condition. The line y = mx + c passes through (-16, 0). So 0 = -16m + c, which gives c = 16m.
3. Tangency condition. The circle has centre (-10, 0) and radius 2. Write the line as mx - y + c = 0. Its perpendicular distance from the centre must equal the radius.

   (|-10m + 16m|)/√(1+m²) = 2 ⇒ 36m² = 4 + 4m² ⇒ m² = 1/8

4. Get m and c. Since m > 0, m = 1/2√2. Then c = 16m = 8/√2 = 4√2.
5. So. 4√2(m + c) = 4√2 × 1/2√2 + 4√2 × 4√2 = 2 + 32 = 34.

The idea to keep. A focal chord always passes through the focus, and that one fact links c to m. For tangency to a circle, set the distance from the centre equal to the radius.

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