Question

find the ends of the major axis and foci of this ellipse.
\\( \frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 16 } = 1 \\)
major axis : \\( ( pm square, 0 ) \\)
foci : \\( ( pm sqrt { square }, 0 ) \\)

Explanation:

Step1: Identify \(a\) and \(b\)

For the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) (\(a>b\)), from \(\frac{x^{2}}{81}+\frac{y^{2}}{16}=1\), we have \(a^{2}=81\), \(b^{2} = 16\). Then \(a=\sqrt{81}=9\), \(b=\sqrt{16} = 4\).

Step2: Find the ends of the major axis

The ends of the major axis for the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b\)) are \((\pm a,0)\). Since \(a = 9\), the ends of the major axis are \((\pm9,0)\).

Step3: Calculate \(c\)

Use the formula \(c^{2}=a^{2}-b^{2}\). Substitute \(a^{2}=81\) and \(b^{2}=16\) into it, we get \(c^{2}=81 - 16=65\).

Answer:

The ends of the major axis: \(9\); Foci: \(65\)