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  • The angle between a pair of tangents drawn from a point P to the parabola   is 45&

The angle between a pair of tangents drawn from a point P to the parabola  y^2=4 a x is 45°. Show that the locus of the point P is a hyperbola. 

Option: 1

Hyperbola


Option: 2

Parabola


Option: 3

Straight Line

 


Option: 4

Ellipse 


Answers (1)

best_answer

Let P (α,β) be any point on the locus. Equation of pair of tangents from P (α ,β) to the parabola,

\begin{aligned} & y^2=4 a x \text { is } \\ & {[\beta y-2 a(x+\alpha)]^2=\left(\beta^2-4 a \alpha\right)\left(y^2-4 a \alpha\right)} \\ & \Rightarrow \beta^2 y^2+4 a^2\left(x^2+\alpha^2+2 x . \alpha\right)-4 a \beta y(x+\alpha)=\beta^2 y^2-4 \beta^2 a x-4 a \alpha y^2+16 a^2 \alpha x \\ & \quad \Rightarrow \beta^2 y^2+4 a^2 x^2+4 a^2 \alpha^2+8 a^2 x . \alpha=\beta^2 y^2-4 \beta^2 a x-4 a \alpha y^2-16 a^2 \alpha x- \\ & 4 a \beta x y-4 a \beta \alpha y \end{aligned}

coefficient of \begin{aligned} & x^2=4 a^2 \\ \end{aligned}

coefficient of x y=-4 a \beta \\

coefficient of y^2=4 a \alpha

Again, angle between the two of Eq. (i) is given as 45° 

\begin{aligned} & \therefore \tan 45^{\circ}=\frac{2 \sqrt{h^2-a b}}{a+b} \\ & \Rightarrow 1=\frac{2 \sqrt{h^2-a b}}{a+b} \end{aligned}

\begin{aligned} & \Rightarrow a+b=2 \sqrt{h^2-a b} \\ & \Rightarrow(a+b)^2=2\left(h^2-a b\right) \\ & \Rightarrow(4 a+4 a \alpha)^2=4\left[4 a^2 \beta^2-\left(4 a^2\right)(4 a \alpha)\right] \\ & \Rightarrow(4 a+4 a \alpha)^2=4\left[4 a^2 \beta^2-\left(4 a^2\right)(4 a \alpha)\right] \\ & \Rightarrow(\alpha+3 a)^2-\beta^2=8 a^2 \end{aligned}

Thus, the required equation of the locus is (x+3 a)^2-y^2=8 a^2which is a hyperbola. 

 

Posted by

Gaurav

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