about to only mathematics

Options

A. 3
B. 6
C. 9
D. 15

Answer

D

Explanation

PLAN (i) `y=mx+a//m` is an equation of tangent to the parabola `y^(2)=4ax.`
(ii) A line is a tangent to circle, if distance of line from centre is equal to the radius of circle.
(iii) Equation of chord drawn from exterior point `(x_(1),y_(1))` to a circle/parabola is given by T=0.
(iv) Area of trapezium` =(1)/(2) ` (Sum of parallel sides)
Let equation of tangent to parabola be ` y= mx+ (2)/(m) `
It also touches the circle ` x^(2)+y^(2)=2.`
`therefore |(2)/(msqrt(1+m^(2)))|=sqrt(2)`
` rArr m^(4)+m^(2)=2 rArr m^(4)+m^(2)-2=0`
` rArr (m^(2)-1)(m^(2)+2)=0 `
` rArr m= pm 1, m^(2)=-2 " " ["rejected " m^(2)=-2] `
So, tangents are `y= x+2, y= -x-2.`
They, intersect at (-2,0).

Equation of chord PQ is ` -2x=2 rArr x=-1`
Equation of chord RS is ` O=4(x-2) rArr x=2 `
` therefore ` Coordinates of P,Q,R,S are
` P(-1,1),Q(-1,-1),R(2,4),S(2,-4) `
` therefore ` Area of quadrilateral `=((2+8)xx3)/(2)=15` sq units