Two tangents are drawn to the hyperbola `x^2/a^2-y^2/b^2=1` such that product of their slope is `c^2` . the locus of the point of intersection is

Options

A. `x^(2)-a^(2)=c^(2)(y^(2)+b^(2))`
B. `x^(2)+a^(2)=c^(2)(y^(2)-b^(2))`
C. `y^(2)+b^(2)=c^(2)(x^(2)-a^(2))`
D. None of these

Answer

3

Explanation

Equation of tangent in the slope form is
`y=mx pm sqrt(a^(2)m^(2)-b^(2))`
Let the tangents be drawn from (h,K)
`thereforek=mh" "sqrt(a^(2)m^(2)-b^(2))`
`rArr(h^(2)-a^(2)m^(2)-2hkm+(k^(2)+b^(2))=0-(i)`
Let it roots are `m_(1)" and "m_(2)`
`rArr m_(1)m_(2)=a^(2)`
`rArr(k^(2)+b^(2))/(h^(2)-a^(2))e^(2)`
`therefore"locus"(x^(2)-y^(2))c^(2)=y^(2)=b^(2)`