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  • P is a point. Two tangents are drawn from it to the parabola such that the slope of one tangent is thr

P is a point. Two tangents are drawn from it to the parabola y^2=4 x such that the slope of one tangent is three times the slope of the other. The locus of P is

Option: 1

a straight line 


Option: 2

a circle
 


Option: 3

a parabola


Option: 4

an ellipse


Answers (1)

best_answer

(c) Let P=(\alpha, \beta) Any tangent to the parabola is

                             y=m x+\frac{a}{m}

It passes through (\alpha, \beta) . So, 

                   \beta=m \alpha+\frac{1}{m} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \quad(\because \text { here, } a=1)

\therefore \: \: \: \: \: \: \: \: \quad m^2 \alpha-\beta m+1=0

It roots are m_1 \text { and } 3 m_1

So,                m_1+3 m_1=\frac{\beta}{\alpha} \text { and } m_1 \cdot m_1=\frac{1}{\alpha}

\therefore \: \: \: \: \: \: \: \: \: \: \: \: \: \quad 3 \cdot\left(\frac{\beta}{4 \alpha}\right)^2=\frac{1}{\alpha} \Rightarrow 3 \beta^2=16 \alpha

Thus, the locus is 3 y^2=16 x, which is a parabola.

Posted by

Ajit Kumar Dubey

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