Given : A circle, `2x^2+""2y^2=""5` and a parabola, `y^2=""4sqrt(5)""x` . Statement - I : An equation of a common tangent to these curves is `y="x+"sqrt(5)` Statement - II : If the line, `y=m x+(sqrt(5))/m(m!=0)` is their common tangent, then m satisfies `m^4-3m^2+""2""=0.` (1) Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I (2) Statement -I is True; Statement -II is False. (3) Statement -I is False; Statement -II is True (4) Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I

Options

A. Statement I is correct, Statement II is correct, Statement II is a correct explanation for statement I
B. Statement I is correct, Statement II is correct, Statement II is not a correct explanation for statement I
C. Statement I is correct, Statement II is incorrect
D. Statement I is incorrect, Statement II is correct

Answer

A

Explanation

Equation of circle can be rewriteen as `x^(2) + y^(2) = (5)/(2)`
Centre `to` (0,0) and radius `to sqrt((5)/(2))`
Let common tangent be
`y = mx + (sqrt(5))/(m) implies m^(2) x - my + sqrt(5) = 0`
The perpendicular from centre to the tangenet is equal to radius of the circle.
`:. (sqrt(5)//m)/(sqrt(1 + m^(2))) = sqrt((5)/(2))`
`implies m sqrt(1 + m^(2)) = sqrt(2)`
`implies m^(2) (1 + m^(2)) = 2`
`implies (m^(2) + 2) (m^(2) = 1) = 0`
`implies m = +- 1` `[:' m^(2) + 2 =! 2 =!` as `m in R]`
`:. y = +- (x + sqrt(5))`, both statements are correct as `m +- 1`
satisfies the given equation of statement II.