Let the tangents drawn to the circle, x 2 + y 2 = 16 from the point P 0 , h meet the x -axis at points A and B . If the area of Δ A P B is minimum, then positive value of h is:

Step-by-step solution

Let equation of tangent be x cos ⁡   α + y sin ⁡   α = 4 Now, its points of intersection with co-ordinate axis are A 4 cosα , 0 ,   P 0 , 4 sinα Now, ar ∆ PAB = 2 ar ∆ PAO = 2 × 1 2 OA × OP = 4 sin ⁡ α     .   4 cos ⁡ α = 32 sin ⁡ 2 α ≥ 32 which is maximum if, α = π 4 ∴       P 0 , 4 sin ⁡ π 4 ∴       h = 4 2