Let A, B and C be three points on the parabola and let the line segment AB meet the line L through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and B on L. then is equal to __________.

Question

Let A, B and C be three points on the parabola $y^{2} = 6 x$ and let the line segment AB meet the line L through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and B on L. then ${(\frac{A M \cdot B N}{C D})}^{2}$ is equal to __________. [2024]

Smart Achievers · Accepted Answer

(36)

Equation of parabola, $y^{2} = 6 x$

$i . e ., y^{2} = 4 \times \frac{3}{2} x$

Let $A (\frac{3}{2} t_{1}^{2}, 3 t_{1}), B (\frac{3}{2} t_{2}^{2}, 3 t_{2}) and C (\frac{3}{2} t_{3}^{2}, 3 t_{3})$

be points on parabola $y^{2} = 6 x$ .

Equation of AB is given by

$\Rightarrow (y - 3 t_{1}) = \frac{3 t_{2} - 3 t_{1}}{\frac{3}{2} (t_{2}^{2} - t_{1}^{2})} (x - \frac{3}{2} t_{1}^{2})$

$\Rightarrow y - 3 t_{1} = \frac{2}{t_{1} + t_{2}} \frac{(2 x - 3 t_{1}^{2})}{2}$

$\Rightarrow (y - 3 t_{1}) (t_{1} + t_{2}) = 2 x - 3 t_{1}^{2}$

$\Rightarrow y (t_{1} + t_{2}) - 3 t_{1}^{2} - 3 t_{1} t_{2} = 2 x - 3 t_{1}^{2}$

$\Rightarrow y (t_{1} + t_{2}) = 2 x + 3 t_{1} t_{2}$

For point D, $y = 3 t_{3}$

$\Rightarrow 2 x = 3 t_{1} t_{3} + 3 t_{2} t_{3} - 3 t_{1} t_{2}$

$\Rightarrow x = \frac{3}{2} (t_{1} t_{3} + t_{2} t_{3} - t_{1} t_{2})$

$| C D | = \frac{3}{2} (t_{1} t_{3} + t_{2} t_{3} - t_{1} t_{2}) - \frac{3}{2} t_{3}^{2}$

$| A M | = | 3 t_{1} - 3 t_{3} |, | B N | = | 3 t_{2} - 3 t_{3} |$

So, ${(\frac{A M \cdot B N}{C D})}^{2} = {(\frac{9 (t_{1} - t_{3}) (t_{2} - t_{3})}{\frac{3}{2} (t_{1} - t_{3}) (t_{3} - t_{2})})}^{2} = {(- 6)}^{2} = 36$ .

Solution

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Solution

Q. Let A, B and C be three points on the parabola y2=6x and let the line segment AB meet the line L through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and B on L. then (AM·BNCD)2 is equal to __________. [2024]

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