If the tangent at the point $P (x_{1}, y_{1})$ to the parabola $y^{2} = 4 a x$ meets the parabola $y^{2} = 4 a (x + b)$ at $Q$ & $R,$ then the mid point of $Q R$ is

(x_{1} + b, y_{1} + b)

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(x_{1} - b, y_{1}, - b)

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(x_{1}, y_{1})

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(x_{1} + b, y_{1})

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Solution

The correct option is C $(x_{1}, y_{1})$
Equation of tangent to $y^{2} = 4 a x$ at $p (x_{1}, y_{1})$ is
$y y_{1} = 2 a (x + x_{1})$
$\Rightarrow 2 a x - y y_{1} + 2 a x_{1} = 0$ ..... (i)
Let $(h, k)$ be mid point of chord $Q R .$
Then equation of $Q R$ is
$k y - 2 a (x + h) - 4 a b = k^{2} - 4 a (h + b)$
$\Rightarrow - 2 a x + k y + 2 a h - k^{2} = 0$ ..... (ii)
Clearly (i) and (ii) represents same line.
$\frac{2 a}{- 2 a} = \frac{- y_{1}}{k} = \frac{2 a x_{1}}{2 a h - k^{2}}$
$y_{1} = k$ and $2 a x_{1} = k^{2} - 2 a h$
$2 a x_{1} = y_{1}^{2} - 2 a h$
$2 a x_{1} = 4 a x_{1} - 2 a h \Rightarrow x_{1} = h$
$∴$ mid point of QR is $(x_{1}, y_{1})$

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Similar questions

P (x_{1}, y_{1})

y^{2} = 4 a x

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If the tangents to the parabola $y_{2} = 4 a x$ at the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ meet at the point $(x_{3}, y_{3})$ , then

The equation of the tangent at $(x_{1}, y_{1})$ to a curve $y^{2} = 4 a x$ at a point $(x_{1}, y_{1})$ is given by $y y_{1} = 2 a (x + x_{1})$

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