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

Studdy Solution

STEP 1

Assumptions1. The given equation is of an ellipse. The general equation of an ellipse is $\frac{x^{}}{a^{}}+\frac{y^{}}{b^{}}=1$ , where $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis.

STEP 2

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

STEP 3

To find the lengths of the semi-major axis and the semi-minor axis, we take the square root of $a^{2}$ and $b^{2}$ respectively.
For $a$ , we have $a = \sqrt{25}$

STEP 4

Calculate the value of $a$ .
$a = \sqrt{25} =$

STEP 5

For $b$ , we have $b = \sqrt{9}$

STEP 6

Calculate the value of $b$ .
$b = \sqrt{9} =3$

STEP 7

Now that we have the values of $a$ and $b$ , we can write the general equation of the ellipse as $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

STEP 8

Substitute the values of $a$ and $b$ into the equation.
$\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$

STEP 9

implify the equation.
$\frac{x^{2}}{25}+\frac{y^{2}}{9}=$ This is the general equation of the ellipse.

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

Studdy Solution

STEP 1

Assumptions1. The given equation is of an ellipse. The general equation of an ellipse is $\frac{x^{}}{a^{}}+\frac{y^{}}{b^{}}=1$ , where $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis.

STEP 2

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

STEP 3

To find the lengths of the semi-major axis and the semi-minor axis, we take the square root of $a^{2}$ and $b^{2}$ respectively.
For $a$ , we have $a = \sqrt{25}$

STEP 4

Calculate the value of $a$ .
$a = \sqrt{25} =$

STEP 5

For $b$ , we have $b = \sqrt{9}$

STEP 6

Calculate the value of $b$ .
$b = \sqrt{9} =3$

STEP 7

Now that we have the values of $a$ and $b$ , we can write the general equation of the ellipse as $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

STEP 8

Substitute the values of $a$ and $b$ into the equation.
$\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$

STEP 9

implify the equation.
$\frac{x^{2}}{25}+\frac{y^{2}}{9}=$ This is the general equation of the ellipse.

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QuestionDetermine the general equation of the ellipse: x29+y225=1\frac{x^{2}}{9}+\frac{y^{2}}{25}=19x2​+25y2​=1.

Studdy Solution

STEP 1

Assumptions1. The given equation is of an ellipse. The general equation of an ellipse is xa+yb=1\frac{x^{}}{a^{}}+\frac{y^{}}{b^{}}=1ax​+by​=1, where aaa is the length of the semi-major axis and bbb is the length of the semi-minor axis.

STEP 2

In the given equation, x29+y225=1\frac{x^{2}}{9}+\frac{y^{2}}{25}=19x2​+25y2​=1, we can see that a2=25a^{2}=25a2=25 and b2=9b^{2}=9b2=9.

STEP 3

To find the lengths of the semi-major axis and the semi-minor axis, we take the square root of a2a^{2}a2 and b2b^{2}b2 respectively.For aaa, we havea=25a = \sqrt{25}a=25​

STEP 4

Calculate the value of aaa.a=25=a = \sqrt{25} =a=25​=

STEP 5

For bbb, we haveb=9b = \sqrt{9}b=9​

STEP 6

Calculate the value of bbb.b=9=3b = \sqrt{9} =3b=9​=3

STEP 7

Now that we have the values of aaa and bbb, we can write the general equation of the ellipse asx2a2+y2b2=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1a2x2​+b2y2​=1

STEP 8

Substitute the values of aaa and bbb into the equation.x252+y232=1\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=152x2​+32y2​=1

STEP 9

implify the equation.x225+y29=\frac{x^{2}}{25}+\frac{y^{2}}{9}=25x2​+9y2​=This is the general equation of the ellipse.

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

Assumptions1. The given equation is of an ellipse. The general equation of an ellipse is $\frac{x^{}}{a^{}}+\frac{y^{}}{b^{}}=1$ , where $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis.

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

To find the lengths of the semi-major axis and the semi-minor axis, we take the square root of $a^{2}$ and $b^{2}$ respectively.
For $a$ , we have $a = \sqrt{25}$

Calculate the value of $a$ .
$a = \sqrt{25} =$

For $b$ , we have $b = \sqrt{9}$

Calculate the value of $b$ .
$b = \sqrt{9} =3$

Now that we have the values of $a$ and $b$ , we can write the general equation of the ellipse as $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

Substitute the values of $a$ and $b$ into the equation.
$\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$

implify the equation.
$\frac{x^{2}}{25}+\frac{y^{2}}{9}=$ This is the general equation of the ellipse.