The equation of chord to the parabola $y^{2} = 4 x$ whose sum of ordinates and product of abscissas of the endpoints of the chord is $4$ and $9$ respectively, is :

2 x - 2 y + 6 = 0

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2 x - 2 y - 6 = 0

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3 x - 2 y + 3 = 0

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3 x - 2 y - 3 = 0

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Solution

The correct options are
A $2 x - 2 y + 6 = 0$
B $2 x - 2 y - 6 = 0$
Let the endpoints of the chord are $A (a t_{1}^{2}, 2 a t_{1})$ and $B (a t_{2}^{2}, 2 a t_{2})$
Where $a = 1$ , so
$A = (t_{1}^{2}, 2 t_{1}), B = (t_{2}^{2}, 2 t_{2})$
Now,
$2 t_{1} + 2 t_{2} = 4 \Rightarrow t_{1} + t_{2} = 2 \dots (1)$
Also,
$t_{1}^{2} \times t_{2}^{2} = 9 \Rightarrow t_{1} t_{2} = \pm 3 \dots (2)$

Equation of the chord is given by
$2 x - y (t_{1} + t_{2}) + 2 a t_{1} t_{2} = 0$
From equation $(1)$ and $(2)$ , we get
$2 x - 2 y \pm 6 = 0$

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