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  • If the tangent at the point to the parabola <img alt="\mathrm{y^2=8 x}" src="https://

If the tangent at the point \mathrm{P(2,4)} to the parabola \mathrm{y^2=8 x} meets the parabola \mathrm{y^2=8 x+5} at Q and R, then the midpoint of Q R is
 

Option: 1

 (2,4)


Option: 2

 (4,-2)


Option: 3

 (7,9)

 


Option: 4

 none of these


Answers (1)

best_answer

 The equation of the tangent to \mathrm{ y^2=8 x} at \mathrm{ P(2,4)} is

\mathrm{4 y=4(x+2) \text { or } x-y+2=0 }
Let \mathrm{\left(x_1, y_1\right)}  be the mid-point of chord Q R. Then, equation of Q R is
\begin{aligned} & \mathrm{ y y_1-4\left(x+x_1\right)-5=y_1^2-8 x_1-5} \\ & \mathrm{4 x-y y_1-4 x_1+y_1^2=0} \end{aligned}
Clearly, (i) and (ii) represent the same line.

So, \mathrm{\frac{4}{1}=-\frac{y_1}{-1}=\frac{-4 x_1+y_1^2}{2} }
\mathrm{\Rightarrow \quad y_1=4 \text { and } 8=4 x_1+y_1^2}

 \mathrm{\Rightarrow y_1=4 \text { and } x_1=2 \text {. }}

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shivangi.shekhar

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