Question

Graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16

Accepted Answer

Identify the standard form of the ellipse equation. The given equation is $$ (x + 1)^2$ + 4(y + 4)^2$ = 100 . To $$write it in the standard form of an ellipse, divide both sides by 100 to get $$ \frac{(x + 1)^2$}{100} + \frac{4(y + 4)^2$}{100} = 1 . $$Simplify the second term by dividing 4 by 100: $$ \frac{(x + 1)^2$}{100} + \frac{(y + 4)^2$}{25} = 1 . $$Now the equation is in the form $$ \frac{(x - h)^2$}{a^2$} + \frac{(y - k)^2$}{b^2$} = 1 $$, where the center is at $$(-1, -4). $$Determine the values of $$a^2$$ and $$b^2. $$Here, $$a^2 = 100$ and b^2$ = 25. $$Since $$a^2$$ \u003e $$b^2$$, the major axis is horizontal, and the ellipse stretches more along the x-axis. Find the lengths of the semi-major axis $$a$$ and semi-minor axis $$b by $$taking the square roots: $$a = \sqrt{100} = 10$$ and $$b = \sqrt{25} = 5. $$Calculate the focal distance $$c$$ using the relationship $$c^2$$ = $$a^2$$ - $$b^2. $$Then, find the coordinates of the foci by moving $$c$$ units left and right from the center along the major axis (x-axis). The foci are at $$(-1 - c, -4)$$ and $$(-1 + c, -4).$$

Graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16

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Key Concepts

Standard Form of an Ellipse

Identifying the Center and Axes Lengths

Finding the Foci of an Ellipse