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Parametric Equation of Parabola

If t 1 and t ...

Question

If $t_{1} a n d t_{2}$ are end of a focal chord of the parabola $(y^{2} = 4 a x)$ then $t_{1} . t_{2} = - 1$

A

True

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B

False

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Solution

The correct option is A
True

Given parabola B,

$y^{2} = 4 a x$

focus=(a,0)

ends of focal chords $A \equiv (a t_{1}^{2}, 2 a t_{1})$

$B \equiv (a t_{1}^{2}, 2 a t_{2})$

AB is given by the equation,

$2 x - (t_{1} + t_{2}) y + 2 a t_{1} t_{2} = 0$

This passes throug (a,0) since chord is focal chort,

$2 a - 0 + 2 a t_{1} t_{2} = 0$

i.e., $2 a t_{1} t_{2} = 0$

$∴ t_{1} t_{2} = - 1$

A natural corollary of this is that if one points is $(a t^{2}, 2 a t)$

then other will be $(\frac{a}{t^{2}}, - \frac{2 a}{t})$

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t_{1}

t_{2}

are the extremities of any focal chord of the parabola

y^{2} = 4 a x

t_{1} t_{2} =

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Q. If the coordinates of the Parabola of the ends of a focal chord of Parabola

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