Problem 99AE

Parametric equations for an ellipse Consider the parametric equations

where a, b, c, and d are real numbers

a. Show that (apart from a set of special cases) the equations describe an ellipse of the form Ax2 + Bxy + Cy2 = K, where A, B, C, and K are constants

b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the x- and y-axes provided ab + cd = 0.

c. Show that the equations describe a circle provided ab + cd = 0 and c2 + d2 = a2 + b2 ≠ 0.

Solution 99AE

a. Now squaring and adding the equations (3) and (4) we get

(dx - by) 2 + (cx - ay)2 = (ad - bc)2

x2 (c2 +d2) + y2 (a2 + b2) -2xy (ac +bd) = (ad - bc) 2

This is in the form Ax2 + Cy2 +Bxy = k

Here B2 - 4AC = 4 (ac + bd)2 -4(c2 +d2) (a2 + b2)

= 4[2abcd- b2c2 -a2d2]

= -4(bc + ad)2

<0

(5) represents an ellipse

b) Take the coefficient of xy is equal to zero in (5) that is ac + bd = 0 we get

x2 (c2 +d2) + y2 (a2 + b2) = (ad - bc) 2

+ =1

It is an ellipse with its axes is x, y

c) We know that Ax2 + Cy2 + Bxy = k represents a circle if B= 0 ,A =C

Equation (5) represents a circle if ac+bd = 0 and a2 + b2 = c2 + d2 0