Question

The circumference of a circle passing through the point \( (4, 6) \) with two normals represented by \( 2x - 3y + 4 = 0 \) and \( x + y - 3 = 0 \) is:

• A. \( 5\pi \)
• B. \( 10\pi \)
• C. \( 25\pi \)
• D. \( 8\pi \)

Hint:

In problems involving the circumference of a circle passing through a point with given normals, use the standard formula for the distance between two lines and then calculate the radius.

Answer 1

The Correct Option is B

To determine the circumference of the circle passing through the point \( (4, 6) \) with two normals given by \( 2x
- 3y + 4 = 0 \) and \( x + y
- 3 = 0 \), follow these steps: Step 1: Understand the problem
- A normal to a circle is a line that passes through the center of the circle.
- The two given lines \( 2x
- 3y + 4 = 0 \) and \( x + y
- 3 = 0 \) are normals to the circle, so they intersect at the center of the circle.
- The circle passes through the point \( (4, 6) \), which lies on its circumference. Step 2: Find the center of the circle The center of the circle is the point of intersection of the two normals. Solve the system of equations: \[ 2x
- 3y + 4 = 0 \quad \text{(1)}
x + y
- 3 = 0 \quad \quad \text{(2)} \] From equation (2), express \( y \) in terms of \( x \): \[ y = 3
- x \] Substitute \( y = 3
- x \) into equation (1): \[ 2x
- 3(3
- x) + 4 = 0
2x
- 9 + 3x + 4 = 0
5x
- 5 = 0
x = 1 \] Substitute \( x = 1 \) into \( y = 3
- x \): \[ y = 3
- 1 = 2 \] Thus, the center of the circle is \( (1, 2) \). Step 3: Find the radius of the circle The radius \( r \) is the distance between the center \( (1, 2) \) and the point \( (4, 6) \) on the circumference. Use the distance formula: \[ r = \sqrt{(4
- 1)^2 + (6
- 2)^2}
r = \sqrt{3^2 + 4^2}
r = \sqrt{9 + 16}
r = \sqrt{25}
r = 5 \] Step 4: Compute the circumference The circumference \( C \) of a circle is given by: \[ C = 2\pi r \] Substitute \( r = 5 \): \[ C = 2\pi \cdot 5 = 10\pi \] Final Answer: The circumference of the circle is \( 10\pi \). \[ \boxed{10\pi} \]