Find the equation of a circle which touches the X-axis and whose centre is `(asintheta,acostheta)`.

Answer

To find the equation of a circle that touches the X-axis and has its center at the point \((A \sin \theta, A \cos \theta)\), we can follow these steps:

### Step 1: Understand the general equation of a circle
The general equation of a circle with center \((x_1, y_1)\) and radius \(r\) is given by:
\[ (x - x_1)^2 + (y - y_1)^2 = r^2 \]

### Step 2: Substitute the center of the circle
Given the center of the circle is \((A \sin \theta, A \cos \theta)\), we substitute \(x_1 = A \sin \theta\) and \(y_1 = A \cos \theta\) into the general equation:
\[ (x - A \sin \theta)^2 + (y - A \cos \theta)^2 = r^2 \]

### Step 3: Determine the radius \(r\)
Since the circle touches the X-axis, the distance from the center to the X-axis must be equal to the radius \(r\). The distance from the center \((A \sin \theta, A \cos \theta)\) to the X-axis is simply the y-coordinate of the center, which is \(A \cos \theta\). Therefore, we have:
\[ r = A \cos \theta \]

### Step 4: Substitute the radius into the equation
Now, we substitute \(r\) into the equation of the circle:
\[ (x - A \sin \theta)^2 + (y - A \cos \theta)^2 = (A \cos \theta)^2 \]

### Step 5: Write the final equation
This gives us the final equation of the circle:
\[ (x - A \sin \theta)^2 + (y - A \cos \theta)^2 = A^2 \cos^2 \theta \]

### Summary
The equation of the circle that touches the X-axis and has its center at \((A \sin \theta, A \cos \theta)\) is:
\[ (x - A \sin \theta)^2 + (y - A \cos \theta)^2 = A^2 \cos^2 \theta \]