Two tangent lines l₁ and l₂ are drawn from the point (2,0) to the parabola 2 y²=-x. If the lines l₁ and l₂ are also tangent to the circle (x-5)²+y²=r, then 17r is equal to ___________.

Solution

Maths sir · 5 steps

Find both tangents from (2,0) in slope form, then the circle's radius is their distance from (5,0).

1. Standard form. Write the parabola as y² = -1/2x. Compare with y² = 4ax: 4a = -1/2, so a = -1/8.
2. Tangent through (2,0). A tangent of slope m is y = mx + a/m = mx - 1/8m. Put in (2,0): 0 = 2m - 1/8m, so 16m² = 1 and m = ±1/4.
3. The two lines. With m = ±1/4, the tangents are y = ±1/4(x - 2), that is, x - 4y - 2 = 0 and x + 4y - 2 = 0.
4. Distance from centre. The circle has centre (5,0) and radius √r. Its distance from either line is (|5 - 0 - 2|)/√(1 + 16) = 3/√17. Both lines give the same distance, so one circle touches both.

   √r = 3/√17 ⇒ r = 9/17

5. So. 17r = 17 × 9/17 = 9.

The idea to keep. For tangents from an external point, put the point into y = mx + a/m and solve for m. Watch the sign of a, and read the circle's right side as (radius)².

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