Question

What is the shape of the graph of the following parametric equations?

x = 3 cos t and y = 2 sin t + 1
(1 point)
Responses

circle
circle

ellipse
ellipse

hyperbola
hyperbola

parabola
parabola

Answer 1

To analyze the shape of the graph given by the parametric equations \( x = 3 \cos t \) and \( y = 2 \sin t + 1 \), we can eliminate the parameter \( t \).

From the equation for \( x \):
\[
\cos t = \frac{x}{3}
\]

From the equation for \( y \):
\[
\sin t = \frac{y - 1}{2}
\]

Now, we use the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \):
\[
\left(\frac{y - 1}{2}\right)^2 + \left(\frac{x}{3}\right)^2 = 1
\]

Expanding this equation gives us:
\[
\frac{(y - 1)^2}{4} + \frac{x^2}{9} = 1
\]

To express it in standard form, we can multiply through by 36 (the least common multiple of 4 and 9):
\[
9(y - 1)^2 + 4x^2 = 36
\]

Rearranging gives:
\[
\frac{(y - 1)^2}{4} + \frac{x^2}{9} = 1
\]

This equation represents an ellipse centered at \( (0, 1) \) with semi-major axis 3 (along the x-axis) and semi-minor axis 2 (along the y-axis).

Therefore, the correct response is:
**ellipse**.