1.

x+y=2 and x-y=2 are tangents on a parabola at (1,1) and (4,2) respectivley. Which of the followings is/are correct.

Answer»

EQUATION of directrix is `x+3y=2`
Equation of axis is `3x-y=5`
Focus of the parabola is at `(8/5,6/5)`
Vertex of the parabola is at `(33/20,13/20)`

Solution :`:.` Tangents are `_|_r`. So , they intersect on directrix.
Point of intesection `=(2,0)` mid-point of `(1,1)` & `(4,2)` is `(5/2,3/2)`
Slope of axis`=(3/2-0)/(5/2-2)=3`
Equation of directrix `y=-1/3(x-2)`
Equation of directrix `y=-1/3(x-2)`
`x+3y=2`
`AB` is focal CHORD,
`BS=` (`_|__(R)` distance from `B` on directrix) `=(4+6-2)/(sqrt(10))=8/(sqrt(10))`
`AS=` ( `_|__(r)` distance from `A` on directrix) `=(1+3-2)/(sqrt(10))=2/(sqrt(10))`
So, focus divides `AB` in `1:4` RATIOS. So `S=(8/5, 6/5)`


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