$$ \Rightarrow xy = {{105} \over {64}}$$ which is the required locus.
2
AIEEE 2005
MCQ (Single Correct Answer)
The locus of a point $$P\left( {\alpha ,\beta } \right)$$ moving under the condition that the line $$y = \alpha x + \beta $$ is tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ is
A
an ellipse
B
a circle
C
a parabola
D
a hyperbola
Explanation
Tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ is
$$y = mx \pm \sqrt {{a^2}{m^2} - {b^2}} $$
Given that $$y = \alpha x + \beta $$ is the tangent of hyperbola
$$ \Rightarrow m = \alpha $$ and $${a^2}{m^2} - {b^2} = {\beta ^2}$$