The locus of a point, from where the tangents to the rectangular hyperbola `x^2-y^2=a^2` contain an angle of `45^0` , is `(x^2+y^2)^2+a^2(x^2-y^2)=4a^2` `2(x^2+y^2)^2+4a^2(x^2-y^(2))=4a^2` `(x^2+y^2)^2+4a^2(x^2-y^2)=4a^2` `(x^2+y^2)+a^2(x^2-y^(2))=a^4`

Options

A. `(x^(2)+y^(2))^(2)+a^(2)(x^(2)-y^(2))=4a^(2)`
B. `2(x^(2)+y^(2))^(2)+4a^(2)(x^(2)-y^(2))=4a^(2)`
C. `(x^(2)+y^(2))^(2)+4a^(2)(x^(2)-y^(2))=4a^(4)`
D. `(x^(2)+y^(2))^(2)+a^(2)(x^(2)-y^(2))=a^(4)`

Answer

C

Explanation

Let `y=mx pm sqrt(m^(2)a^(2)-a^(2))` be two tangents that pass through (h,k). Then
`(k-mh)^(2)=m^(2)a^(2)-a^(2)`
`"or "m^(2)(h^(2)-a^(2))-2khm+k^(2)+a^(2)=0`
`"or "m_(1)+m_(2)=(2kh)/(h^(2)-a^(2))`
and `m_(1)m_(2)=(k^(2)+a^(2))/(h^(2)-a^(2))`
Now, `tan45^(@)=(m_(1)-m_(2))/(1+m_(1)m_(2))`
`"or "1=((m_(1)+m_(2))^(2)-4m_(1)m_(2))/((1+m_(1)m_(2))^(2))`
`"or "(1+(k^(2)+a^(2))/(h^(2)-a^(2)))^(2)=((2kh)/(h^(2)-a^(2)))^(2)-4((k^(2)+a^(2))/(h^(2)-a^(2)))`
`"or "(h^(2)+k^(2))^(2)=4h^(2)k^(2)-4(k^(2)+a^(2))(h^(2)-a^(2))`
`"or "(x^(2)+y^(2))^(2)=4(a^(2)y^(2)-a^(2)x^(2)+a^(4))`
`"or "(x^(2)+y^(2))^(2)+4a^(2)(x^(2)-y^(2))=4a^(4)`