QuestionDetermine the general equation of the ellipse from $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is in the standard form of an ellipse $\frac{x^{}}{a^{}}+\frac{y^{}}{b^{}}=1$ . . The ellipse is centered at the origin (0,0).
3. The values of a and b are the square roots of the denominators in the equation, representing the semi-major and semi-minor axes respectively.

STEP 2

Identify the values of a and b from the given equation.In the given equation $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$ , we can see that $a^{2}=9$ and $b^{2}=25$ .

STEP 3

Calculate the values of a and b by taking the square root of $a^{2}$ and $b^{2}$ .
$a=\sqrt{9}$ $b=\sqrt{25}$

STEP 4

Calculate the values of a and b.
$a=\sqrt{9}=3$ $b=\sqrt{25}=$

STEP 5

The general form of an ellipse equation is $Ax^{2} + By^{2} + Cx + Dy + =0$ .To convert the standard form to the general form, we need to multiply both sides of the equation by $a^{2}b^{2}$ .
$a^{2}b^{2}(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}})=a^{2}b^{2}$

STEP 6

Substitute the values of a and b into the equation.
$3^{2}5^{2}(\frac{x^{2}}{3^{2}}+\frac{y^{2}}{5^{2}})=3^{2}5^{2}$

STEP 7

implify the equation.
$225(x^{2}+y^{2})=225$

STEP 8

Multiply out the left side of the equation.
$225x^{2}+225y^{2}=225$

STEP 9

Subtract225 from both sides of the equation to get the general form of the ellipse equation.
$225x^{2}+225y^{2}-225=$ So, the general equation of the ellipse given the standard form equation $\frac{x^{2}}{9}+\frac{y^{2}}{25}=$ is $225x^{2}+225y^{2}-225=$ .

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