An ellipse has equation (x-1)²/9 + (y+2)²/25 = 1. In which direction does the major axis run?
AHorizontally, because the x-term appears first in the equation
BHorizontally, because 9 is the denominator under x
CVertically, because 25 > 9 and 25 is under the y-term
DVertically, because the center has a negative y-coordinate
The major axis runs in the direction of the larger denominator. Here a² = 25 (under the y-term), so a = 5 and the major axis is vertical — vertices are 5 units above and below the center (1, -2). The common mistake is assuming a² always goes under x; the rule is simply that whichever variable has the larger denominator determines the direction of the major axis, regardless of which variable appears first.
Question 2 Multiple Choice
An ellipse has a = 5 and b = 3. How far from the center are the foci located?
A√34, because c² = a² + b² = 25 + 9
B4, because c² = a² − b² = 25 − 9 = 16
C8, because c = a + b
D2, because c = a − b
For an ellipse, c² = a² − b² (not a² + b²; that formula belongs to hyperbolas). So c² = 25 − 9 = 16, giving c = 4. The foci are 4 units from the center along the major axis, inside the ellipse — not at the vertices. The most dangerous distractor uses c² = a² + b², which is the hyperbola relationship. Remembering the geometric derivation (a right triangle at the co-vertex endpoint with legs b and c, hypotenuse a) makes the ellipse formula unforgettable.
Question 3 True / False
A circle is a special case of an ellipse in which both foci coincide at the center, because when a = b the value of c equals zero.
TTrue
FFalse
Answer: True
When a = b, c² = a² − b² = 0, so c = 0 — both foci are at the same point (the center). With both foci at the center, the sum-of-distances definition reduces to twice the distance to the center, which is constant — exactly the definition of a circle. So every circle is an ellipse with eccentricity e = c/a = 0.
Question 4 True / False
The foci of an ellipse are located at the endpoints of the major axis (the vertices).
TTrue
FFalse
Answer: False
The foci are inside the ellipse, between the center and the vertices. The vertices are at distance a from the center; the foci are at distance c, where c² = a² − b² means c < a. Only if b = 0 would c equal a, but that collapses the ellipse to a degenerate line segment. Confusing foci with vertices is the most common geometric misconception about ellipses — remember that the foci are always in the interior.
Question 5 Short Answer
Explain why the relationship between a, b, and c for an ellipse is c² = a² − b², not c² = a² + b². What geometric reasoning produces this formula?
Think about your answer, then reveal below.
Model answer: Consider an endpoint of the minor axis (a co-vertex). By symmetry, the two distances from this point to the two foci are equal. Since the defining sum of distances equals 2a, each distance from a co-vertex to a focus must equal a. This forms a right triangle with the focus, center, and co-vertex as vertices: the legs are c (center to focus) and b (center to co-vertex), and the hypotenuse is a (focus to co-vertex). By the Pythagorean theorem, a² = b² + c², which rearranges to c² = a² − b².
The formula c² = a² + b² (where b would be the hypotenuse) belongs to hyperbolas. For ellipses, a is always the largest parameter — it is the hypotenuse — because c < a and b < a. Keeping the co-vertex right-triangle picture in mind distinguishes the two cases and makes the formula self-deriving.