Math / Geometry

Question12/27
Find the value of "C" (distance of focus) in the given ellipse 2.24
5 13 3.60 Alina Thomas

Studdy Solution

STEP 1

1. The ellipse is centered at the origin.
2. The major axis is along the x-axis.
3. The endpoints of the major axis are at $x = -5$ and $x = 5$ .
4. The endpoints of the minor axis are at $y = -2$ and $y = 2$ .

STEP 2

1. Identify the lengths of the semi-major and semi-minor axes.
2. Use the relationship between the semi-major axis, semi-minor axis, and the distance of the foci to find $C$ .

STEP 3

Identify the lengths of the semi-major and semi-minor axes:
- The semi-major axis length $a$ is half the distance of the major axis: $a = \frac{5 - (-5)}{2} = 5$
- The semi-minor axis length $b$ is half the distance of the minor axis: $b = \frac{2 - (-2)}{2} = 2$

STEP 4

Use the relationship between the semi-major axis, semi-minor axis, and the distance of the foci $C$ :
- The relationship is given by the equation: $c^2 = a^2 - b^2$
- Substitute the known values: $c^2 = 5^2 - 2^2$ $c^2 = 25 - 4$ $c^2 = 21$
- Solve for $c$ : $c = \sqrt{21}$
The distance of the focus $C$ is:
$\boxed{\sqrt{21}}$

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