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**Unformatted text preview: **− c)2 − (x + c)2
4a2 − 4cx
a2 − cx
a2 − cx 2 a4 − 2a2 cx + c2 x2
a4 − 2a2 cx + c2 x2
a4 − a2 c2
a2 a2 − c2 We are nearly ﬁnished. Recall that b2 = a2 − c2 so that
a2 − c2 x2 + a2 y 2 = a2 a2 − c2
b2 x2 + a2 y 2 = a2 b2
x2 y 2
+2 =1
a2
b
1 In other words, tons and tons of Intermediate Algebra. Stay sharp, this is not for the faint of heart. 422 Hooked on Conics This equation is for an ellipse centered at the origin. If the ellipse were centered at a point (h, k ),
we would get the following
Equation 7.4. The Standard Equation of an Ellipse: For positive unequal numbers a and
b, the equation of an ellipse with center (h, k ) is
(x − h)2 (y − k )2
+
=1
a2
b2
Some remarks about Equation 7.4 are in order. First note that the values a and b determine
how far in the x and y directions, respectively, one counts from the center to arrive at points on
the ellipse. Also take note that if a > b, then we have an ellipse whose major axis is horizontal,
and hence, the foci lie to the l...

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