The curve with parametric equations `x=1+4 cos theta, y=2+3 sin theta ` is

Options

A. an ellipse
B. a parabola
C. a hyperbola
D. a circle

Answer

To determine the type of curve represented by the parametric equations \( x = 1 + 4 \cos \theta \) and \( y = 2 + 3 \sin \theta \), we can follow these steps:

### Step 1: Rearrange the equations for \( \cos \theta \) and \( \sin \theta \)
From the equation for \( x \):
\[ x - 1 = 4 \cos \theta \]
This gives:
\[ \cos \theta = \frac{x - 1}{4} \]

From the equation for \( y \):
\[ y - 2 = 3 \sin \theta \]
This gives:
\[ \sin \theta = \frac{y - 2}{3} \]

### Step 2: Use the Pythagorean identity
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the expressions for \( \sin \theta \) and \( \cos \theta \) into this identity:
\[ \left(\frac{y - 2}{3}\right)^2 + \left(\frac{x - 1}{4}\right)^2 = 1 \]

### Step 3: Simplify the equation
Expanding both terms:
\[ \frac{(y - 2)^2}{9} + \frac{(x - 1)^2}{16} = 1 \]

### Step 4: Identify the type of curve
The equation we have now is in the standard form of an ellipse:
\[ \frac{(x - 1)^2}{16} + \frac{(y - 2)^2}{9} = 1 \]
This confirms that the curve is indeed an ellipse.

### Conclusion
Thus, the curve represented by the given parametric equations is an ellipse.

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