The curve represented by `x=2(cost+sint) and y = 5(cos t-sin t )` is

Options

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

Explanation

To determine the type of curve represented by the parametric equations \( x = 2(\cos t + \sin t) \) and \( y = 5(\cos t - \sin t) \), we will follow these steps:

### Step 1: Rewrite the equations
We start with the given parametric equations:
\[ x = 2(\cos t + \sin t) \]
\[ y = 5(\cos t - \sin t) \]

### Step 2: Express \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\)
From the equation for \(x\):
\[ \frac{x}{2} = \cos t + \sin t \]
From the equation for \(y\):
\[ \frac{y}{5} = \cos t - \sin t \]

### Step 3: Solve for \(\cos t\) and \(\sin t\)
Let:
\[ u = \cos t \quad \text{and} \quad v = \sin t \]
We can rewrite the equations as:
\[ u + v = \frac{x}{2} \quad (1) \]
\[ u - v = \frac{y}{5} \quad (2) \]

### Step 4: Add and subtract the equations
Adding equations (1) and (2):
\[ (u + v) + (u - v) = \frac{x}{2} + \frac{y}{5} \]
This simplifies to:
\[ 2u = \frac{x}{2} + \frac{y}{5} \quad \Rightarrow \quad u = \frac{x}{4} + \frac{y}{10} \]

Subtracting equation (2) from (1):
\[ (u + v) - (u - v) = \frac{x}{2} - \frac{y}{5} \]
This simplifies to:
\[ 2v = \frac{x}{2} - \frac{y}{5} \quad \Rightarrow \quad v = \frac{x}{4} - \frac{y}{10} \]

### Step 5: Use the Pythagorean identity
We know that:
\[ u^2 + v^2 = 1 \]
Substituting \(u\) and \(v\):
\[ \left(\frac{x}{4} + \frac{y}{10}\right)^2 + \left(\frac{x}{4} - \frac{y}{10}\right)^2 = 1 \]

### Step 6: Expand and simplify
Expanding both squares:
\[ \left(\frac{x}{4}\right)^2 + 2\left(\frac{x}{4}\right)\left(\frac{y}{10}\right) + \left(\frac{y}{10}\right)^2 + \left(\frac{x}{4}\right)^2 - 2\left(\frac{x}{4}\right)\left(\frac{y}{10}\right) + \left(\frac{y}{10}\right)^2 = 1 \]
This simplifies to:
\[ 2\left(\frac{x}{4}\right)^2 + 2\left(\frac{y}{10}\right)^2 = 1 \]
\[ \frac{x^2}{8} + \frac{y^2}{50} = 1 \]

### Step 7: Identify the conic section
The equation \(\frac{x^2}{8} + \frac{y^2}{50} = 1\) is in the standard form of an ellipse, where \(a^2 = 50\) and \(b^2 = 8\).

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

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