Problem 43
Question
Graph each ellipse and give the location of its foci. $$\frac{x^{2}}{25}+\frac{(y-2)^{2}}{36}=1$$
Step-by-Step Solution
Verified Answer
The graph of the given ellipse is centered at (0, 2), with its major axis along the y-axis of lengths 12 units and minor axis along the x-axis of lengths 10 units. The foci are at (0, 2- sqrt(11)) and (0, 2+ sqrt(11)).
1Step 1 - Identify the centroid
In this equation, x is not translated and y is translated by 2 units upwards, so the centroid is at the coordinates (0, 2).
2Step 2 - Draw the axes and ellipse
The lengths of the semi axes are given by the square roots of the numbers under x and y. So, draw the major and minor axes of the ellipse according to the given lengths (6 units in the y direction and 5 units in the x direction). Then sketch the ellipse around these axes.
3Step 3 - Find the length of the focal distance (c)
Apply the formula c = sqrt(b^2 - a^2) . Here a = 5, b = 6, so we get c = sqrt( 36 - 25) = sqrt(11) units.
4Step 4 - Find the coordinates of the foci
An ellipse with a vertical major axis has its foci at (h, k ± c), so in this case, the coordinates of the two foci are (0, 2±sqrt(11)).
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