Question

use the information provided to write the standard form equation of each ellipse.

1. vertices: (0, 6), (0, -6)

co - vertices: (5, 0), (-5, 0)

use the information provided to write the standard form equation of each circle.

1. center: (13, 6)

radius: 2

Explanation:

Problem 23 (Ellipse)

Step1: Identify major axis orientation

Vertices are on the y - axis (\((0,6)\), \((0, - 6)\)), so major axis is vertical. Co - vertices are on the x - axis (\((5,0)\), \((-5,0)\)). The standard form of an ellipse with vertical major axis is \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\) (\(a>b>0\)), where \(a\) is the distance from center to vertex, \(b\) is the distance from center to co - vertex.
Center is the mid - point of vertices (or co - vertices). Mid - point of \((0,6)\) and \((0,-6)\) is \((0,0)\), so center \((h,k)=(0,0)\).
\(a=\vert6 - 0\vert = 6\), \(b=\vert5 - 0\vert = 5\).

Step2: Write the equation

Substitute \(h = 0\), \(k = 0\), \(a = 6\), \(b = 5\) into the standard form \(\frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}}=1\).
We get \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{6^{2}}=1\), i.e., \(\frac{x^{2}}{25}+\frac{y^{2}}{36}=1\).

Problem 24 (Circle)

Step1: Recall circle's standard form

The standard form of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^{2}+(y - k)^{2}=r^{2}\).

Step2: Substitute values

Given center \((h,k)=(13,6)\) and radius \(r = 2\). Substitute \(h = 13\), \(k = 6\), \(r = 2\) into the formula: \((x - 13)^{2}+(y - 6)^{2}=2^{2}\).

Answer:

s:

1. \(\boldsymbol{\frac{x^{2}}{25}+\frac{y^{2}}{36}=1}\)
2. \(\boldsymbol{(x - 13)^{2}+(y - 6)^{2}=4}\)