Question

Equation of family of ellipse having foci on x axis and center at the origin

Answer 1

The line segment of length 2b perpendicular to the transverse axis whose midpoint is the center is the conjugate axis of the hyperbola. The standard equation for a hyperbola with a vertical transverse axis is - = 1. The center is at (h, k). The distance between the vertices is 2a.

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Answer 2

Toolbox:

Family of ellipses with foci on x-axis and centre at the origin x2a2+y2b2=1

Step 1:

Family of ellipses with foci on x-axis and centre at the origin x2a2+y2b2=1

Differentiating with respect to x we get,

2xa2+2yb2dydx=0

⇒yx(dydx)=−b2a2

Step 2:

Differentiating with respect to x we get,

yx(d2ydx2)+xdy/dx−yx2dydx=0

xyd2ydx2+x(dydx)2−ydydx=0

Which is required differential equation.