Question

If the parametric equations of the circle passing through the points (3,4), (3,2) and (1,4) is x = a + r cosθ, y = b + r sinθ then b^a r^a = 

• A. 9
• B. 18
• C. 27
• D. 54

Answer 1

The Correct Option is B

To find the value of \(b^a r^a\) for the circle passing through the points \((3, 4)\), \((3, 2)\), and \((1, 4)\), where \((a, b)\) is the center and \(r\) is the radius, we proceed as follows:

1. Setting Up the Circle’s Equation:
The general equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). Since the circle passes through \((3, 4)\), \((3, 2)\), and \((1, 4)\), we write:

\( (3 - h)^2 + (4 - k)^2 = r^2 \quad (1) \)
\( (3 - h)^2 + (2 - k)^2 = r^2 \quad (2) \)
\( (1 - h)^2 + (4 - k)^2 = r^2 \quad (3) \)

2. Finding the Center’s y-Coordinate:
Equate equations (1) and (2) since both equal \(r^2\):

\( (3 - h)^2 + (4 - k)^2 = (3 - h)^2 + (2 - k)^2 \)
The \((3 - h)^2\) terms cancel, leaving:

\( (4 - k)^2 = (2 - k)^2 \)
Expand both sides:

\( 16 - 8k + k^2 = 4 - 4k + k^2 \)
Subtract \(k^2\) from both sides:

\( 16 - 8k = 4 - 4k \)
\( 12 = 4k \implies k = 3 \)
So, the y-coordinate of the center is \(k = 3\).

3. Finding the Center’s x-Coordinate:
Equate equations (1) and (3):

\( (3 - h)^2 + (4 - 3)^2 = (1 - h)^2 + (4 - 3)^2 \)
Since \(4 - 3 = 1\), both \((4 - 3)^2 = 1\), so:

\( (3 - h)^2 = (1 - h)^2 \)
Expand:

\( 9 - 6h + h^2 = 1 - 2h + h^2 \)
Subtract \(h^2\) from both sides:

\( 9 - 6h = 1 - 2h \)
\( 8 = 4h \implies h = 2 \)
So, the x-coordinate of the center is \(h = 2\). The center is \((2, 3)\).

4. Calculating the Radius:
Use equation (1) to find \(r^2\):

\( (3 - 2)^2 + (4 - 3)^2 = r^2 \)
\( 1^2 + 1^2 = 1 + 1 = 2 \)
\( r = \sqrt{2} \)

5. Identifying \(a\), \(b\), and Computing \(b^a r^a\):
The center is \((a, b) = (2, 3)\), so \(a = 2\), \(b = 3\). The radius is \(r = \sqrt{2}\). Compute:

\( b^a r^a = 3^2 (\sqrt{2})^2 = 9 \cdot 2 = 18 \)

Final Answer:
The value of \(b^a r^a\) is \(18\).