The circle with equation x^2 + y^2 = 1 is transformed to an ellipse through dilations of factor 4 from the x-axis and factor 3 from the y-axis, and a translation of 4 units in the positive x-direction and 3 units in the positive y-direction. Find the equation of the ellipse.
HINT: Ellipses have the form (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
HINT: Ellipses have the form (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
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You can pretty much write the equation directly from the data given.
((x - 4)^2 / 16) + ((y - 3)^2 / 9) = 1
The (x-4) and (y-3) terms correspond to the translation, and the /16 and /9 divisions correspond to the two dilations of the original unit circle.
Just as a check, the new ellipse is supposed to have a semi major axis length of 4 in the X direction (it used to be 1 for the original unit circle, but we dilated by a factor of four in X), and center at (4,3). Thus, if we offset this center by the semi major length in X, the point (8,3) should satisfy the equation, and sure enough:
(8-4)^2 / 16 + 0 = 16/16 = 1
Ahhh, unless there's a wording subtlety here, and dilation FROM the X-axis is not the same as dilation ALONG the X-axis. In that case, swap the "9" and "16" in the two denominators.
((x - 4)^2 / 16) + ((y - 3)^2 / 9) = 1
The (x-4) and (y-3) terms correspond to the translation, and the /16 and /9 divisions correspond to the two dilations of the original unit circle.
Just as a check, the new ellipse is supposed to have a semi major axis length of 4 in the X direction (it used to be 1 for the original unit circle, but we dilated by a factor of four in X), and center at (4,3). Thus, if we offset this center by the semi major length in X, the point (8,3) should satisfy the equation, and sure enough:
(8-4)^2 / 16 + 0 = 16/16 = 1
Ahhh, unless there's a wording subtlety here, and dilation FROM the X-axis is not the same as dilation ALONG the X-axis. In that case, swap the "9" and "16" in the two denominators.