Statement 1: An equation of a common tangent to the parabola `y^2=16sqrt(3)x` and the ellipse `2x^2+""y^2=""4 ""i s ""y""=""2x""+""2sqrt(3)` . Statement 2: If the line `y""=""m x""+(4sqrt(3))/m ,(m!=0)` is a common tangent to the parabola `y^2=""16sqrt(3)x` and the ellipse `2x^2+""y^2=""4` , then m satisfies `m^4+""2m^2=""24` .

Options

A. Statement 1 is true , statement 2 is true , statement 2 is a correct explanation for statement 1.
B. Statement 1 is true , statement 2 is true , statement 2 is not correct explanation for statement 1.
C. Statement 1 is true , statement 2 is false.
D. Statement 1 is false , statement 2 is true.

Answer

B

Explanation

2 Tangent to parabola `y^(2)=4sqrt(5)x`, having slope m is :
`y=mx+(sqrt(5))/(m)`
This line is tangent to the circle `2x^(2)+2y^(2)=5`
`rArr` Distance of center from the line is equal to its radius.
`rArr" "|{:(mxx0-0+(sqrt(5))/(m))/(sqrt(1+m^(2))):}|=sqrt((5)/(2))`
`rArr" "m^(4)+m^(2)-2=0rArrm^(2)=1rArrm=pm1`
Also `m^(2)=1` satisfies `m^(4)-3m^(2)+2=0`
One of the common tangents for m=1 is `y=x+sqrt(5)`.
Thus, both statements are correct, statement 2 is not correct explanation of statement 1