Let P be the point of intersections of the common tangents to the parabola `y^(2)=12x` and the hyperbola ` 8x^(2) -y^(2)=8.` If S and S' denotes the foci of the hyperbola where S lies on the positive X-axis then P divides SS' in a ratio.

Options

A. `13:11`
B. `14:13`
C. `5:4`
D. `2:1`

Answer

C

Explanation

Equation of given parabola `y^(2)=12x " ...(i)" `
and hyperbola `8x^(2) -y^(2)=8 " ...(ii) " `
Now, equation of tangent to parabola `y^(2)=12x ` having slope 'm' is `y = mx+(3)/(m) " ...(iii)" `
and equation of tagent to hyperbola
`(x^(2))/(1)-(y^(2))/(8)=1` having slope 'm' is
`y=mx pm sqrt(1^(2)m^(2)-8) " ...(iv)" `
Since, tagents (iii) and (iv) represent the same line
` therefore m^(2) -8=((3)/(m))^(2)`
`rArr m^(4)-8m^(2)-9=0`
`rArr (m^(2)-9)(m^(2)+1)=0`
`rArr m =pm 3.`
Now, equation of common tangents to the parabola (i) and hyperbola (ii) are `y = 3x +1 and y = - 3x - 1`
` because ` Point 'P' is point of intersection of above common tangents,
` therefore P(-1//3,0)`
and focus of hyperbola S(3, 0) and S'(-3, 0).
Thus, the required ratio `=(PS)/(PS')=(3+1//3)/(3-1//3)=(10)/(8)=(5)/(4)`