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If the tangent at the point P\left(x_1, y_1\right) to the parabola y^2=4 ax meets the parabola y^2=4 a(x+b) at Q and R, then the mid-point of QR is

 

Option: 1

(0,1)


Option: 2

(2,4)


Option: 3

(x_{1},y_{1})

 


Option: 4

(-1,0)


Answers (1)

best_answer

Equation of the tangent at P\left(x_1, y_1\right) \text { to } y^2=4 a x  is 

                          y y_1-2 a x-2 a x_1=0

Equation of the chord of y^2=4 a(x+b) whose mid point is 

\begin{gathered} \left(x^{\prime}, y^{\prime}\right) \text { is } y y^{\prime}-2 a x-2 a x^{\prime}-4 a b \\ \\=y^{\prime 2}-4 a x^{\prime}-4 a b \end{gathered}

\text { (ie) } y y^{\prime}-2 a x-\left(y^{\prime 2}-2 a x^{\prime}\right)=0\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ....(ii)

Eqs. (i) and (ii) represent the same line 

 \therefore \quad \frac{y_1}{y^{\prime}}=\frac{2 a}{2 a}=-\frac{2 a x_1}{y^{\prime 2}-2 a x^{\prime}}

This gives y^{\prime}=y_1, and then 2 a x_1=y^{\prime 2}-2 a x^{\prime}

\begin{array}{ll} \Rightarrow & =y_1^2-2 a x^{\prime}=4 a x_1-2 a x \\ \\\therefore & x^{\prime}=x_1 \end{array}

\therefore \text { Mid point }\left(x^{\prime}, y^{\prime}\right)=\left(x_1, y_1\right)

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