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  • Let a curve <img alt="\mathrm{y=f\left ( x \right ),x\: \varepsilon \left ( 0,\infty \right )}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7By%3Df%5Cleft%20%28%20x%20%5Crig

Let a curve \mathrm{y=f\left ( x \right ),x\: \varepsilon \left ( 0,\infty \right )} pass through the points \mathrm{P\left ( 1,\frac{3}{2} \right )} and \mathrm{Q\left ( a,\frac{1}{2} \right )} If the tangent at any point
\mathrm{R (b,f(b))} to the given curve cuts the y-axis at the points \mathrm{S(0, c) } such that \mathrm{bc = 3}, then \mathrm{\left ( PQ \right )^{2}} is equal to _____:

Option: 1

5


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

Equation of tangent at \mathrm{R(b, f(2)) is \: y-f(b)=f^{\prime}(b) \cdot(x-b)}
which passes through (0, c)
\begin{aligned} & \Rightarrow \mathrm{c}-\mathrm{f}(\mathrm{b})=\mathrm{f}^{\prime}(\mathrm{b}) \cdot(-\mathrm{b}) \\ & \Rightarrow \frac{3}{\mathrm{~b}}-\mathrm{f}(\mathrm{b})=\mathrm{f}(\mathrm{b}) \cdot(-\mathrm{b}) \\ & \Rightarrow \frac{\mathrm{bf} \mathrm{f}^{\prime}(\mathrm{b})-\mathrm{f}(\mathrm{b})}{\mathrm{b}^2}=-\frac{3}{\mathrm{~b}^3} \\ & \Rightarrow d\left(\frac{\mathrm{f}(\mathrm{b})}{\mathrm{b}}\right)=-\frac{3}{\mathrm{~b}^3} \Rightarrow \frac{\mathrm{f}(\mathrm{b})}{\mathrm{b}}=\frac{3}{2 b^2}+\lambda \end{aligned}
Which passes through (1,3 / 2)
\begin{aligned} & \Rightarrow \frac{3}{2}=\frac{3}{2}+\lambda \Rightarrow \lambda=0 \\ & \Rightarrow \mathrm{f}(\mathrm{b})=\frac{3}{2 \mathrm{~b}} \\ & \mathrm{f}(\mathrm{a})=\frac{1}{2} \Rightarrow \frac{1}{2}=\frac{3}{2 \mathrm{~b}} \Rightarrow \mathrm{b}=3 \\ & \Rightarrow \mathrm{c}=1 \Rightarrow \mathrm{Q}(3,1 / 2) \\ & \Rightarrow \mathrm{PQ}^2=2^2+(1)^2=5 \end{aligned}

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Kuldeep Maurya

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