Question

# If the point (λ,λ+1) lies inside the region bounded by the curve x=√25−y2 and y-axis, then λ belongs to the interval

A

(1,3)

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B

(4,3)

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C

(,4)(3,)

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D

none of these

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Solution

## The correct option is A (−1,3) The given equation of the curve is x=√25−y2Squaring on both sides, we get ⇒x2=25−y2⇒x2+y2=25Since (λ,λ+1) lies inside the region bounded by the curve x2+y2=25 and the y-axis, we have: λ2+(λ+1)2<25, provided λ+1>0 ⇒λ2+λ2+1+2λ<25,λ>−1 [Since, (a+b)2=a2+2ab+b2] ⇒2λ2+2λ−24<0, λ>−1 ⇒λ2+λ−12<0, λ>−1 ⇒(λ−3)(λ+4)<0, λ>−1 ⇒−4<λ<3, λ>−1⇒λ∈(−4,3) and λ∈(−1,∞) ∴λ∈(−1,3)

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