Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Write the equation in standard form for the ellipse $x^2 + 2y^2 - 6x - 23 = 0$ .

View step-by-step help

View step-by-step help

Convert equations of ellipses from general to standard form

Full solution

Q. Write the equation in standard form for the ellipse

x^2 + 2y^2 - 6x - 23 = 0

.

Complete the Square for $x$ : Now, we complete the square for the $x$ terms. To do this, we take the coefficient of the $x$ term, divide it by $2$ , and square it. That's $(-6/2)^2 = 9$ . Add $9$ to both sides. $\newline$ $x^2 - 6x + 9 + 2y^2 = 23 + 9$
Factor out $2$ for $y$ : We do the same for the $y$ terms, but since there's a coefficient of $2$ in front of $y^2$ , we need to factor that out first. $\newline$ $2(y^2) = 2(y^2 + 0y + 0)$ $\newline$ We don't need to complete the square for $y$ because there's no $y$ term to complete the square with. $\newline$ So, we just have $2(y^2)$ .
Rewrite with Completed Square: Now we rewrite the equation with the completed square for $x$ and the $y$ term. $\newline$ $(x - 3)^2 + 2(y^2) = 32$
Divide by $32$ : Divide everything by $32$ to get the standard form of the ellipse. $(x - 3)^2/32 + 2(y^2)/32 = 1$
Simplify $y$ Term: Simplify the $y$ term by dividing the coefficient $2$ by $32$ . $\left(x - 3\right)^2/32 + \left(y^2\right)/16 = 1$

More problems from Convert equations of ellipses from general to standard form

Question

Write the equation in standard form for the ellipse with center at the origin, vertex

(-10,0)

, and co-vertex

(0,3)

Posted 1 month ago

Question

Write the equation in standard form for the ellipse

7x^2 + y^2 = 21

\newline

Posted 2 months ago

Question

What is the center of the ellipse

x^2 + 2y^2 - 16 = 0

\newline

Write your answer in simplified, rationalized form.

\newline

(\_\_\_\_\_,\_\_\_\_\_)

Posted 1 month ago

Question

What is the length of the major axis of the ellipse

\frac{x^2}{9} + \frac{y^2}{2} = 1

\newline

Write your answer in simplified, rationalized form.

\newline

\_\_\_\_\_

Posted 1 month ago

Question

What is the center of the ellipse

\left(\frac{(x - 5)^2}{9}\right) + \left(\frac{(y - 2)^2}{54}\right) = 1

\newline

Write your answer in simplified, rationalized form.

\newline

(________ , _______)

Posted 2 months ago

Question

The equation of an ellipse is given below.

\newline

\frac{(x+15)^{2}}{676}+\frac{(y-4)^{2}}{100}=1

\newline

What are the foci of this ellipse?

\newline

1

\newline

(-15+\sqrt{24}, 4)

(-15-\sqrt{24}, 4)

\newline

(-15,4+\sqrt{24})

(-15,4-\sqrt{24})

\newline

(-39,4)

(9,4)

\newline

(-15,28)

(-15,-20)

Posted 3 months ago

Question

The equation of an ellipse is given below.

\newline

\frac{(x-5)^{2}}{3}+\frac{(y-7)^{2}}{6}=1

\newline

What are the foci of this ellipse?

\newline

1

\newline

(5,7+\sqrt{3})

(5,7-\sqrt{3})

\newline

(-5,-7+\sqrt{3})

(-5,-7-\sqrt{3})

\newline

(-5+\sqrt{3},-7)

(-5-\sqrt{3},-7)

\newline

(5+\sqrt{3}, 7)

(5-\sqrt{3}, 7)

Posted 3 months ago

Question

The equation of an ellipse is given below.

\newline

\frac{x^{2}}{46}+\frac{(y+8)^{2}}{26}=1

\newline

What are the foci of this ellipse?

\newline

1

\newline

(0+\sqrt{20}, 8)

(0-\sqrt{20}, 8)

\newline

(0,8+\sqrt{20})

(0,8-\sqrt{20})

\newline

(0,-8+\sqrt{20})

(0,-8-\sqrt{20})

\newline

(0+\sqrt{20},-8)

(0-\sqrt{20},-8)

Posted 3 months ago

Question

Write an equation for an ellipse centered at the origin, which has foci at

( \pm 12,0)

and vertices at

( \pm 13,0)

Posted 3 months ago

Question

Write an equation for an ellipse centered at the origin, which has foci at

(0, \pm 6)

and vertices at

(0, \pm \sqrt{37})

Posted 3 months ago