Question
Question: The locus of the mid points of the portion of the tangents to the ellipse intercepted between the ax...
The locus of the mid points of the portion of the tangents to the ellipse intercepted between the axes is
A
a2y2+b2x2=4x2y2
B
a2x2+b2y2=4x2y2
C
x2+y2=a2
D
x2+y2=b2
Answer
a2y2+b2x2=4x2y2
Explanation
Solution
Equation of the tangent to the ellipse a2x2+b2y2=1 at (α cos θ, b sin θ) is 2xcosθ+2ysinθ=1
∴ Coordinates of the mid point of the portion
intercepted between the axes is
(2αsecθ,2bcosecθ) ∴ x = 2αsecθ⇒cosθ=2xα
and y = 2bcosec θ ⇒ sinθ = 2yb ∴ 4x2a2+4y2b2=1
Thus, required locus is a2y2+b2x2=4x2y2