The ellipse · Grade 12 · 5 units

In The ellipse (Grade 12 · 5 units) you practise 6 skills: The axes and the foci · The equation of an ellipse · The sum of the distances from the foci · The ellipse as a squashed circle · An ellipse, a line and a circle · Loci that are ellipses. An ellipse is the locus of all the points whose distances from two fixed points, the foci, add up to a fixed number greater than the distance between the foci. That sum equals the length of the long axis. With the centre at the origin and the foci on the horizontal axis, the standard equation is: x squared over the square of half the long axis, plus y squared over the square of half the short axis, equals one. The square of a focus’s distance from the centre equals the difference between the squares of the half axes. When the greater number is under y squared, the long axis is vertical and the foci lie on it.

An ellipse can also come from a circle: multiply the vertical coordinate of every point on the circle by the same positive number less than one. The horizontal half axis stays the radius, and the vertical half axis is the radius times the number. The area is multiplied by the same number, so the area of the ellipse is pi times the product of the half axes. To find the common points of an ellipse and a line, substitute the line’s equation into the ellipse’s and solve a quadratic equation: the discriminant decides whether there are two common points, one or none. To solve the system of an ellipse and a circle, make y squared the subject of one equation and substitute it into the other. Each solution for x gives two points, one point or none, depending on the sign of y squared.

Tip: The sum of the distances from the foci equals the whole long axis, not half of it. Halve it to get the half axis.

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What you learn

The axes and the foci

For example: The ellipse x²/81 + y²/64 = 1 is written in the form x² / a² + y² / b² = 1 , with a and b positive. What is a ?

Answer and solution

The answer: 9

Here a² = 81 , so a = 9 .

The equation of an ellipse

For example: The foci of an ellipse are (3, 0) and (−3, 0) , and the distances of each of its points from the foci add up to 10 . What is the equation of the ellipse?

Answer and solution

The answer: x²/25 + y²/16 = 1

Here 2a = 10 , so a = 5 and b² = 25 − 9 = 16 . The equation is x²/25 + y²/16 = 1 .

The sum of the distances from the foci

For example: The foci of the ellipse x²/25 + y²/16 = 1 are F and G . The point P is on the ellipse, and PF = 8 . What is PG ?

Answer and solution

The answer: 2

Here a = 5 , so PF + PG = 10 . Then PG = 10 − 8 = 2 .

The ellipse as a squashed circle

For example: The y coordinate of every point on the circle x² + y² = 100 is multiplied by the same positive number, which gives the ellipse x²/100 + y²/49 = 1 . What is the number?

Answer and solution

The answer: 7/10

Here a = 10 and b = 7 , so the number is q = 7/10 .

An ellipse, a line and a circle

For example: The ellipse x²/36 + y²/4 = 1 and the line y = −x + 7 are given. How many common points do they have?

Answer and solution

The answer: No common points

Substituting and clearing the fractions gives 40x² − 504x + 1620 = 0 . Its discriminant is −5184 , so there are no common points.

Loci that are ellipses

For example: What is the locus of the points whose distances from the points A(6, −5) and B(0, 5) add up to 15 ?

Answer and solution

The answer: An ellipse

The distance between A and B is less than 15 , and the distances from the two points add up to a constant: the answer is an ellipse.

Not on the site yet

Parts of this topic that are not built yet: Dandelin’s proof (enrichment) · A circle whose centre is not on an axis.

The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).

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