More properties of an Ellipse with center (0,0) and (h,k)

1. ELLIPSE
MORE PROPERTIES
PRECALCULUS
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2. REMEMBER THIS!
An ellipse is the set of all points on the plane,
the sum of whose distances from two fixed
points is a constant.

3. MORE PROPERTIES OF AN ELLIPSE
Some ellipses have their foci aligned
vertically, and some have centers not at the
origin. Their standard equations and
properties are given in the next slide.
where a > b and 𝑐 = 𝑎2 − 𝑏2 .

6. Example 1. Give the coordinates of the center, foci,
vertices, and covertices of the ellipse with the given
equation. Sketch the graph, and include these points.
(a)
(𝑥+3)2
24
+
(𝑦−5)2
49
= 1
(b) 9x2 + 16y2 - 126x + 64y = 71
Example 2. Find the (standard) equation of the ellipse
whose foci are F1(3,0) and F2(3,0), such that for any
point on it, the sum of its distances from the foci is 10.

7. Solution. (1.a) From a2 = 49 and b2 = 24, we have a
= 7, b =2 6 ≈ 4.9, and c = 𝑎2 − 𝑏2 = 5. The ellipse
is vertical.
center: (-3,5)
foci: F1(-3,0), F2(-3,10)
vertices: V1(-3,-2), V2(-3,12)
covertices: W1(−3 − 2 6,5) ≈(7.9,5)
W2(−3 + 2 6,5) ≈(1.9,5)

8. Example 3. The foci of an ellipse are (-3,-6) and
(-3,2). For any point on the ellipse, the sum of its
distances from the foci is 14. Find the standard equation
of the ellipse.
Example 4. A tunnel has the shape of a semi ellipse that
is 15 ft high at the center, and 36 ft across at the base. At
most how high should a passing truck be, if it is 12 ft
wide, for it to be able to fit through the tunnel? Round
off your answer to two decimal places.

9. Try This!
1. Give the coordinates of the center, foci, vertices, and
covertices of the ellipse with the given equation. Sketch
the graph, and include these points.
36x2 + 20y2 -144x + 120y - 396 = 0
2. The arch of a bridge is in the shape of a semiellipse,
with its major axis at the water level. Suppose the arch
is 20 ft high in the middle, and 120 ft across its major
axis. How high above the water level is the arch, at a
point 20 ft from the center (horizontally)?

10. THANK YOU
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@reylkastro2
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