Question

Let the locus of the centre (α, β), β> 0, of the circle which touches the circle x² +(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

The Correct Option is C

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: x² = 4y

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