The correct option is C (x2+y2)2=16(x24−y29)
Given hyperbola is x24−y29=1
Let any point on hyperbola be
P(2secθ,3tanθ)
Given circle is x2+y2=4
Chord of contact of to a given circle from this point is given by T=0
⇒2secθx+3tanθy=4 ⋯(1)
If M(h,k) is the mid point of the chord, then equation of chord of contact with M as its mid point is given by
T=S1⇒x(h)+y(k)=h2+k2 ⋯(2)
Equation (1) and (2) represent the same chord of contact.
Hence, 2secθh=3tanθk=4h2+k2
⇒secθ=4h2(h2+k2), tanθ=4k3(h2+k2)
We know that,
sec2θ−tan2θ=1⇒16h24(h2+k2)2−16k29(h2+k2)2=1⇒h24−k29=(h2+k2)216
Hence, the locus of the mid point M(h,k) of chord of contact is
(x2+y2)2=16(x24−y29)