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  • The locus of point of intersection of tangent drawn at the end point of variable chord of the parabola <img alt="\mathrm{y}^2=4 \mathrm{ax}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5C

The locus of point of intersection of tangent drawn at the end point of variable chord of the parabola \mathrm{y}^2=4 \mathrm{ax}, which subtends a constant angle \mathrm{\tan ^{-1}(\mathrm{~b})} at the vertex of the parabola is \mathrm{4\left(\mathrm{y}^2-\mathrm{k}_1 \mathrm{ax}\right)=\mathrm{b}^2\left(\mathrm{k}_2 \mathrm{a}+\mathrm{x}\right)^2 \quad where \, \, \, \mathrm{k}_1-\mathrm{k}_2=}

Option: 1

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Option: 2

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Option: 3

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Option: 4

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Answers (1)

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Let the point of intersection drawn at the end point of a variable chord of the parabola \mathrm{y^2= 4 a x} is '\mathrm{ R ' (\alpha, \beta).}

So equation of chord of contact \mathrm{P Q \Rightarrow y \beta=2 a(x+\alpha)} Combined equation of straight line \mathrm{O P} and \mathrm{O Q} is

\mathrm{ \mathrm{y}^2-4 a x\left(\frac{\mathrm{y} \beta}{2 \mathrm{a} \alpha}-\frac{\mathrm{x}}{\alpha}\right)=0 }

Since \mathrm{P Q} subtends angle \mathrm{\tan ^{-1}(b)} at O, so

\mathrm{ \begin{aligned} & b=2 \frac{\sqrt{\frac{\beta^2}{\alpha^2}-\frac{4 a}{\alpha}}}{a+\frac{4 a}{\alpha}}=\frac{2 \sqrt{\beta^2-4 a \alpha}}{4 a+\alpha} \\\\ & \Rightarrow 4\left(\beta^2-4 a \alpha\right)=b^2(4 a+\alpha)^2 \\\\ & \text { So required locus is } 4\left(y^2-4 a x\right)=b^2(4 a+x)^2 \end{aligned} }

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Irshad Anwar

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