The locus of a point whose chord of contact with respect to the circle x2+y2=4 is a tangent to the hyperbola x y=1 is a/an

Question

The locus of a point whose chord of contact with respect to the circle x2+y2=4 is a tangent to the hyperbola x y=1 is a/an (A) ellipse (B) circle (C) hyperbola (D) parabola

Tardigrade Admin · Accepted Answer

Let the point be (h, k). Then the equation of the chord of contact is h x+k y=4. Since h x+k y=4 is tangent to x y=1, x((4-h x/k))=1 has two equal roots. Therefore, h x2-4 x+k=0 or h k=4 Thus, the locus of (h, k) is x y=4. Hence, c2=4

Q. The locus of a point whose chord of contact with respect to the circle $x^{2} + y^{2} = 4$ is a tangent to the hyperbola $x y = 1$ is a/an

ellipse

circle

hyperbola

parabola

Let the point be $(h, k)$ .
Then the equation of the chord of contact is $h x + k y = 4$ .
Since $h x + k y = 4$ is tangent to $x y = 1$ ,
$x (\frac{4 - h x}{k}) = 1$
has two equal roots. Therefore,
$h x^{2} - 4 x + k = 0$
or $hk = 4$
Thus, the locus of $(h, k)$ is $x y = 4$ . Hence,
$c^{2} = 4$

Q. The locus of a point whose chord of contact with respect to the circle x2+y2=4 is a tangent to the hyperbola xy=1 is a/an