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Derivation of condition for three ellipse to have a common tangent.

Derivation of condition for three ellipse to have a common tangent.

Grade:12th pass

1 Answers

Vikas TU
14149 Points
6 years ago
Let the equation of 3 ellipse be
x^2 / a^2   +   y^2 / b^2  = 1----------------------------1
x^2 / c^2   +   y^2 / d^2  = 1-----------------------------2
x^2 / e^2   +   y^2 / f^2  = 1------------------------------3
Differentiate 1,2,3 w.r.t x
Differentiation of (1) is
-  b^2 x / a^2 y-----------------------4
Differentiation of (2) is
-  d^2 x / c^2 y-------------------------5
Differentiation of (3) is
-  f^2 x / e^2 y---------------------6
For ellipse to have common tangent 4,5,6 should be equal to each other
Hence
b2 /a2 = d2 / c2 = f2 / e2 .

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