Chapter Twenty-Two — Conic Sections
The Humongous Book of Algebra Problems
Note: Problems 22.23–22.24 refer to the ellipse with center (–1,0), vertical major axis of
length 10, and horizontal minor axis of length 6.
22.24 Write the equation of the ellipse in standard form.
According to Problem 22.22, the standard form of an ellipse with a vertical
major axis is . The center of the ellipse is (–1,0), so
h = –1 and k = 0. The values of a and b are half of the lengths of the major and
minor axes, respectively. Therefore, a = 10 ÷ 2 = 5 and b = 6 ÷ 2 = 3. Substitute h,
k, a, and b into the standard form equation.
Note: Problems 22.25–22.27 refer to the ellipse with equation x
– 6x – 36y + 36 = 0.
22.25 Write the equation in standard form.
Group the x-terms together in a set of parentheses, group the y-terms in a
second set of parentheses, and move the constant to the right side of the
– 6x) + (9y
– 36y) = 0 – 36
In order to complete the square for the expression 9y
– 36y, the coefﬁcient of
the squared term must be 1.
– 6x) + 9(y
– 4y) = –36
Complete the square for each of the parenthetical quantities.
The equation of an ellipse in standard form contains only the constant 1 on
the right side of the equation. Divide the entire equation by 9, the constant
currently right of the equal sign.
beneath the y-
expression when the
major axis is vertical
and appears beneath
the x-expression when
the major axis is
But it’s not,
so factor the
coefcient (9) out of
the square, you
have to add 9 to
and 4 to the y-
the y-parentheses are
multiplied by 9, so you’re
actually adding 9(4), so
make sure to add
9(4) = 36 to the
right side of the