Question

Let the locus of the centre (α, β), β> 0, of the circle which touches the circle x 2 +(y - 1)2 = 1 externally and also touches the x -axis be L. Then the area bounded by L and the line y = 4 is :

• A. $\frac{\sqrt{32}}{3}$
• B. $\frac{40\sqrt2}{3}$
• C. $\frac{64}{3}$
• D. $\frac{32}{3}$

Answer 1

Answer: (C)

$\frac{64}{3}$

Answer 2

The radius of circle S touching the x -axis and center (α, β) is |β|. According to the given conditions
α2 + (β – 1)2 = (|β| + 1)2
α2 + β2 – 2β + 1 = β2 + 1 + 2|β|
α2 = 4β as β> 0
∴ Required louse is L : x 2 = 4 y

The area of the shaded region =2$\int_{0}^{4}$ 2$\sqrt{ydy}$
=4⋅[y^{\frac{3}{2}}$$\frac{3}{2}]04
=$\frac{64}{3}$ square units