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Question
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Solution
Let the equation of ellipse be
`x^2/"a"^2 + y^2/"b"^2` = 1 ......(i)
Since the major axis is twice the minor axis,
2a = 2(2b)
∴ a = 2b ......(ii)
Substituting (ii) in (i), we get
`x^2/(2"b")^2 + y^2/"b"^2` = 1
∴ `x^2/(4"b"^2) + y^2/"b"^2` = 1
∴ x2 + 4y2 = 4b2
Differentiating w.r.t. x, we get
`2x + 8y ("d"y)/("d"x)` = 0
∴ `x + 4y ("d"y)/("d"x)` = 0, where is the required differential equation.
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