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 :
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 :
\(\frac{\sqrt{32}}{3}\)
\(\frac{40\sqrt2}{3}\)
\(\frac{64}{3}\)
\(\frac{32}{3}\)
The Correct Option is C
Solution and Explanation
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
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Concepts Used:
Tangents and Normals
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m×n = -1
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Diagram Explaining Tangents and Normal:
