A parabola touches the lines y= x and y=-x at P (3, 3) and Q (2,-2) respectively. Then

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Equation of directrix is $x + 5 y = 0$

Equation of line through origin and parallel to axis is $x - 5 y = 0$

Focus is $(\frac{30}{13}, \frac{- 6}{13})$

Equation of line through origin and focus is $x + 5 y = 0$

answer is A, C, D.

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Detailed Solution

Midpoint of PQ= $(\frac{5}{2}, \frac{1}{2})$ =R
Slope of OR= $\frac{1}{5}$
Slope of axis = $\frac{1}{5}$
( $∴$ Line joining the point of intersection of tangents at A and B on parabola and mid-point of chord AB is always parallel to axis of parabola)
$\Rightarrow$ Slope of directrix = -5
Eq.of directrix: y=-5x $\Rightarrow$ 5x+y=0
Circle with OP as diameter: $x^{2} + y^{2} - 3 x - 3 y = 0$
Circle with OQ as diameter: $x^{2} + y^{2} - 2 x + 2 y = 0$
Focus is the point of intersection of circles: $S = (\frac{30}{13}, \frac{- 6}{13})$
Eq.of line joining origin to focus is: x+5y=0