Question

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 `4sqrt(2)` (m + c) is equal to ______.

Options

• 31

• 32

• 33

• 34

Answer 1

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 `4sqrt(2)` (m + c) is equal to 34.

Explanation:

Given equation of parabola y² = –64x

Focus = (–16, 0)

Let focal chord y = mx + c

⇒ c = 16 m  ...(i)

If y = mx + c is tangent to (x + 10)² + y² = 4

⇒ y = `"m"(x + 10) ± 2sqrt(1 + "m"^2)`

∴ c = `10"m" ± 2sqrt(1 + "m"^2)`

So, 16m = `10"m" ± 2sqrt(1 + "m"^2)`  ......[From (i)]

⇒ 6m = `±2sqrt(1 + "m"^2)`

⇒ 3m = `±sqrt(1 + "m"^2)`

Squaring both sides,

⇒ 9m² = 1 + m²

⇒ 8m² = 1

⇒ m = `1/(2sqrt(2)), ("m" > 0)` and c = `8/sqrt(2)`

∴ `4sqrt(2)("m" + "c") = 4sqrt(2)(1/(2sqrt(2)) + 8/sqrt(2))`

= `4sqrt(2)(17/(2sqrt(2)))`

= 34