Question

2 multiple answer 1 point which equation(s) of an ellipse has a vertical major axis? select all that apply. \\(\frac{(x-6)^2}{35} + \frac{(y+11)^2}{53} = 1\\) \\(\frac{x^2}{12} + \frac{y^2}{35} = 1\\) \\(\frac{(x-9)^2}{82} + \frac{(y-10)^2}{120} = 1\\) \\(\frac{(x+7)^2}{92} + \frac{(y+16)^2}{88} = 1\\) \\(\frac{x^2}{53} + \frac{y^2}{25} = 1\\)

Explanation:

Step1: Recall Ellipse Standard Form

The standard form of an ellipse with a vertical major axis is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) where \(a > b\) (so the denominator under the \(y\)-term is larger).

Step2: Analyze Each Equation

• First Equation: \(\frac{(x - 6)^2}{35}+\frac{(y + 11)^2}{53}=1\). Denominator of \(y\)-term (\(53\)) > denominator of \(x\)-term (\(35\)). So vertical major axis.
• Second Equation: \(\frac{x^2}{12}+\frac{y^2}{35}=1\). Denominator of \(y\)-term (\(35\)) > denominator of \(x\)-term (\(12\)). Vertical major axis.
• Third Equation: \(\frac{(x - 9)^2}{82}+\frac{(y - 10)^2}{120}=1\). Denominator of \(y\)-term (\(120\)) > denominator of \(x\)-term (\(82\)). Vertical major axis.
• Fourth Equation: \(\frac{(x + 7)^2}{92}+\frac{(y + 16)^2}{88}=1\). Denominator of \(x\)-term (\(92\)) > denominator of \(y\)-term (\(88\)). Horizontal major axis (not vertical).
• Fifth Equation: \(\frac{x^2}{53}+\frac{y^2}{25}=1\). Denominator of \(x\)-term (\(53\)) > denominator of \(y\)-term (\(25\)). Horizontal major axis (not vertical).

Answer:

• \(\boldsymbol{\frac{(x - 6)^2}{35}+\frac{(y + 11)^2}{53}=1}\)
• \(\boldsymbol{\frac{x^2}{12}+\frac{y^2}{35}=1}\)
• \(\boldsymbol{\frac{(x - 9)^2}{82}+\frac{(y - 10)^2}{120}=1}\)