Question

an ellipse has the equation \\(\frac{(x - 3)^2}{3^2} + \frac{(y + 5)^2}{5^2} = 1\\). if the ellipse is shifted two spaces to the left and five spaces up, what is the new equation?\\(\frac{x^2}{3^2} + \frac{y^2}{5^2} = 1\\)\\(\frac{(x + 2)^2}{3^2} + \frac{(y - 5)^2}{5^2} = 1\\)\\(\frac{(x + 3)^2}{3^2} + \frac{(y + 3)^2}{5^2} = 1\\)\\(\frac{(x - 4)^2}{3^2} + \frac{(y + 10)^2}{5^2} = 1\\)

Explanation:

Step1: Apply left shift to x-term

For a left shift of 2 units, replace $x$ with $x+2$ in the x-expression:
$\frac{(x-2 + 2)^2}{3^2} = \frac{x^2}{3^2}$

Step2: Apply up shift to y-term

For an up shift of 5 units, replace $y$ with $y-5$ in the y-expression:
$\frac{(y+5 - 5)^2}{5^2} = \frac{y^2}{5^2}$

Step3: Combine shifted terms

Assemble the new ellipse equation from the shifted terms.

Answer:

$\boldsymbol{\frac{x^2}{3^2} + \frac{y^2}{5^2} = 1}$ (Option 1)