Let any point on the given circle be O(d cos
, d sin
)
and midpoint of chord PQ of ellipse be M(h,k).
Equation of chord of contact PQ with O as an external point is
T = 0 w.r.t. the ellipse x2/a2 + y2/b2 – 1 = 0.
dcos
x/a
2 + dsin
y/b
2 – 1 = 0 i.e.
dcos
x/a
2 + dsin
y/b
2 = 1 ….(1) is the equation of PQ.
Also, considering midpoint M, equation of chord PQ is
T = S1 w.r.t. the ellipse x2/a2 + y2/b2 – 1 = 0.
hx/a
2 + ky/b
2 – 1 = h
2/a
2 + k
2/b
2 – 1 i.e.
hx/a2 + ky/b2 = h2/a2 + k2/b2 ….(2) is the equation of PQ.
Comparing (1) & (2), we get
h/dcos
= k/dsin
= (h
2/a
2 + k
2/b
2) / 1
cos
= h/d(h
2/a
2 + k
2/b
2) & sin
= k/d(h
2/a
2 + k
2/b
2)
[h/d(h
2/a
2 + k
2/b
2)]
2 + [k/d(h
2/a
2 + k
2/b
2)]
2 = 1
h2 + k2 = d2(h2/a2 + k2/b2)2
Replace h by x and k by y.
x2 + y2 = d2(x2/a2 + y2/b2)2 is the required locus.
(You may simplify the above equation to get the desired result.)
Pls approve.
