The line `x+ y +2=0` is a tangent to a parabola at point A, intersect the directrix at B and tangent at vertex at C respectively. The focus of parabola is `S(a, 0)`. Then

Options

A. CS is perpendicular to AB
B. `AC*BC=CS^(2)`
C. `AC*BC=8`
D. AC=BC

Answer

A, B, C

Explanation

1,2,3
For parabola `y^(2)=4ax`, equation of tangent at `A(at^(2),2at)` is
`ty=x+at^(2)`
`:.C-=(0,at),B-=(-a,at-(a)/(t))`
`AC=atsqrt(1+t^(2)),BC=(a)/(t)sqrt(1+t^(2)),(CS)^(2)=a^(2)(1+t^(2))`
`rArrAC*BC=(CS)^(2)`
Slope of `CSxx` Slope of AB `=(0-at)/(a-0)xx(1)/(t)=-1`
Hence, CS is perpendicular to AB
CS = Distance of S (2,0) and x+y+2=0
`=(|2+2|)/(sqrt(2))=2sqrt(2)`
`:.AC*BC=(CS)^(2)=8`