QuestionFind the center, semi-major axis, semi-minor axis, and foci of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is in the standard form of an ellipse's equation, which is $\frac{(x-h)^{}}{a^{}}+\frac{(y-k)^{}}{b^{}}=1$ , where $(h, k)$ is the center of the ellipse, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis. . The given equation is $\frac{x^{}}{9}+\frac{y^{}}{25}=1$ .

STEP 2

The center of the ellipse is at the point $(h, k)$ . In the given equation, $h$ and $k$ are both0 because the equation is in the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ . Therefore, the center of the ellipse is at the origin, or $(0,0)$ .

STEP 3

The length of the semi-major axis is the larger of the two denominators under the square roots in the equation. In this case, the semi-major axis is under the $y^{2}$ term, and its length is the square root of25, or5.

STEP 4

The length of the semi-minor axis is the smaller of the two denominators under the square roots in the equation. In this case, the semi-minor axis is under the $x^{2}$ term, and its length is the square root of9, or3.

STEP 5

The foci of an ellipse are given by the equation $(h \pm c, k)$ , where $c=\sqrt{a^{2}-b^{2}}$ .

STEP 6

Substitute the values of $a$ and $b$ into the equation to find the value of $c$ .
$c=\sqrt{a^{2}-b^{2}}=\sqrt{5^{2}-3^{2}}$

STEP 7

Calculate the value of $c$ .
$c=\sqrt{25-9}=\sqrt{16}=4$

STEP 8

Substitute the values of $h$ , $k$ , and $c$ into the equation for the foci to find the coordinates of the foci.
$oci = (h \pm c, k) = (0 \pm4,0)$

STEP 9

Calculate the coordinates of the foci.
$oci = (-4,) \, and \, (4,)$ The center of the ellipse is at the origin $(,)$ , the length of the semi-major axis is5, the length of the semi-minor axis is3, and the foci are at $(-4,)$ and $(4,)$ .

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QuestionFind the center, semi-major axis, semi-minor axis, and foci of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$ .

Studdy Solution

STEP 1

STEP 2

The center of the ellipse is at the point $(h, k)$ . In the given equation, $h$ and $k$ are both0 because the equation is in the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ . Therefore, the center of the ellipse is at the origin, or $(0,0)$ .

STEP 3

The length of the semi-major axis is the larger of the two denominators under the square roots in the equation. In this case, the semi-major axis is under the $y^{2}$ term, and its length is the square root of25, or5.

STEP 4

The length of the semi-minor axis is the smaller of the two denominators under the square roots in the equation. In this case, the semi-minor axis is under the $x^{2}$ term, and its length is the square root of9, or3.

STEP 5

The foci of an ellipse are given by the equation $(h \pm c, k)$ , where $c=\sqrt{a^{2}-b^{2}}$ .

STEP 6

Substitute the values of $a$ and $b$ into the equation to find the value of $c$ .
$c=\sqrt{a^{2}-b^{2}}=\sqrt{5^{2}-3^{2}}$

STEP 7

Calculate the value of $c$ .
$c=\sqrt{25-9}=\sqrt{16}=4$

STEP 8

Substitute the values of $h$ , $k$ , and $c$ into the equation for the foci to find the coordinates of the foci.
$oci = (h \pm c, k) = (0 \pm4,0)$

STEP 9

Calculate the coordinates of the foci.
$oci = (-4,) \, and \, (4,)$ The center of the ellipse is at the origin $(,)$ , the length of the semi-major axis is5, the length of the semi-minor axis is3, and the foci are at $(-4,)$ and $(4,)$ .

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QuestionFind the center, semi-major axis, semi-minor axis, and foci of the ellipse x29+y225=1\frac{x^{2}}{9}+\frac{y^{2}}{25}=19x2​+25y2​=1.

Studdy Solution

STEP 1

STEP 2

The center of the ellipse is at the point (h,k)(h, k)(h,k). In the given equation, hhh and kkk are both0 because the equation is in the form x2a2+y2b2=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1a2x2​+b2y2​=1. Therefore, the center of the ellipse is at the origin, or (0,0)(0,0)(0,0).

STEP 3

The length of the semi-major axis is the larger of the two denominators under the square roots in the equation. In this case, the semi-major axis is under the y2y^{2}y2 term, and its length is the square root of25, or5.

STEP 4

The length of the semi-minor axis is the smaller of the two denominators under the square roots in the equation. In this case, the semi-minor axis is under the x2x^{2}x2 term, and its length is the square root of9, or3.

STEP 5

The foci of an ellipse are given by the equation (h±c,k)(h \pm c, k)(h±c,k), where c=a2−b2c=\sqrt{a^{2}-b^{2}}c=a2−b2​.

STEP 6

Substitute the values of aaa and bbb into the equation to find the value of ccc.c=a2−b2=52−32c=\sqrt{a^{2}-b^{2}}=\sqrt{5^{2}-3^{2}}c=a2−b2​=52−32​

STEP 7

Calculate the value of ccc.c=25−9=16=4c=\sqrt{25-9}=\sqrt{16}=4c=25−9​=16​=4

STEP 8

Substitute the values of hhh, kkk, and ccc into the equation for the foci to find the coordinates of the foci.oci=(h±c,k)=(0±4,0)oci = (h \pm c, k) = (0 \pm4,0)oci=(h±c,k)=(0±4,0)

STEP 9

Calculate the coordinates of the foci.oci=(−4,) and (4,)oci = (-4,) \, and \, (4,)oci=(−4,)and(4,)The center of the ellipse is at the origin (,)(,)(,), the length of the semi-major axis is5, the length of the semi-minor axis is3, and the foci are at (−4,)(-4,)(−4,) and (4,)(4,)(4,).

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QuestionFind the center, semi-major axis, semi-minor axis, and foci of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$ .

The center of the ellipse is at the point $(h, k)$ . In the given equation, $h$ and $k$ are both0 because the equation is in the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ . Therefore, the center of the ellipse is at the origin, or $(0,0)$ .

The length of the semi-major axis is the larger of the two denominators under the square roots in the equation. In this case, the semi-major axis is under the $y^{2}$ term, and its length is the square root of25, or5.

The length of the semi-minor axis is the smaller of the two denominators under the square roots in the equation. In this case, the semi-minor axis is under the $x^{2}$ term, and its length is the square root of9, or3.

The foci of an ellipse are given by the equation $(h \pm c, k)$ , where $c=\sqrt{a^{2}-b^{2}}$ .

Substitute the values of $a$ and $b$ into the equation to find the value of $c$ .
$c=\sqrt{a^{2}-b^{2}}=\sqrt{5^{2}-3^{2}}$

Calculate the value of $c$ .
$c=\sqrt{25-9}=\sqrt{16}=4$

Substitute the values of $h$ , $k$ , and $c$ into the equation for the foci to find the coordinates of the foci.
$oci = (h \pm c, k) = (0 \pm4,0)$

Calculate the coordinates of the foci.
$oci = (-4,) \, and \, (4,)$ The center of the ellipse is at the origin $(,)$ , the length of the semi-major axis is5, the length of the semi-minor axis is3, and the foci are at $(-4,)$ and $(4,)$ .