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If a parabola touches the lines y = x a...

If a parabola touches the lines `y = x and y =-x` at `P(3,3) and Q(2,-2)` respectively, then

A

focus is `((30)/(13),(-6)/(13))`

B

equation of directrix is `x +5y = 0`

C

equation of line through origin and focus is `x +5y = 0`

D

equation of line through origin and parallel to axis is `x -5y = 0`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
Mid-point of `PQ -= ((5)/(2),(1)/(2))`
Slope of `OP = (1)/(5)`
`rArr` Slope of axis `= (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]
`rArr` Slope of directrix = -5
So, Equation of dirextrix is `y =- 5x rArr 5x +y =0`. Circle with OP as diameter is `x^(2) + y^(2) - 3x -3y =0`.
Circle with OQ as diameter is `x^(2) + y^(2) -2x +2y =0`. These circles pass through fixed point focus which is focus
`((30)/(13),(-6)/(13))` which we get by solving above two equations. Hence equation of line joining origin to focus is `x + 5y =0`.
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