Question:
Find the equation of the circle touching the x-axis at \( (9, 0) \) and the line \( y = 14 \).
Find the equation of the circle touching the x-axis at \( (9, 0) \) and the line \( y = 14 \).
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When a circle touches the x-axis or a line, use the distance from the center to the axis or line to find the radius and the center. Then use the standard form of the circle's equation.
Updated On: Apr 24, 2025
- \( (x - 9)^2 + y^2 = 14^2 \)
- \( (x - 9)^2 + (y - 7)^2 = 7^2 \)
- \( (x + 9)^2 + (y - 14)^2 = 14^2 \)
- \( (x - 9)^2 + (y - 7)^2 = 14^2 \)
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The Correct Option is B
Solution and Explanation
Equation of a circle that touches the x-axis at \( (9, 0) \) and the line \( y = 14 \).
1. Step 1: Understand the general equation of a circle: The general equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center of the circle, and \( r \) is the radius.
2. Step 2: Use the information about the x-axis contact: Since the circle touches the x-axis at \( (9, 0) \), the radius of the circle is the distance from the center \( (h, k) \) to the x-axis, which is equal to \( k \) (the y-coordinate of the center).
3. Step 3: Use the information about the line \( y = 14 \): The distance from the center of the circle to the line \( y = 14 \) must also be the radius. The distance from a point \( (h, k) \) to a line \( y = c \) is \( |k - c| \). Therefore, the radius \( r = |k - 14| \).
4. Step 4: Set up the system of equations: Since the radius is both \( k \) and \( |k - 14| \), we equate these two expressions: \[ k = |k - 14| \] Solving this gives \( k = 7 \).
5. Step 5: Find the center and radius: The center of the circle is \( (9, 7) \), and the radius is \( 7 \).
6. Step 6: Write the equation of the circle: The equation of the circle is: \[ (x - 9)^2 + (y - 7)^2 = 7^2 \]
1. Step 1: Understand the general equation of a circle: The general equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center of the circle, and \( r \) is the radius.
2. Step 2: Use the information about the x-axis contact: Since the circle touches the x-axis at \( (9, 0) \), the radius of the circle is the distance from the center \( (h, k) \) to the x-axis, which is equal to \( k \) (the y-coordinate of the center).
3. Step 3: Use the information about the line \( y = 14 \): The distance from the center of the circle to the line \( y = 14 \) must also be the radius. The distance from a point \( (h, k) \) to a line \( y = c \) is \( |k - c| \). Therefore, the radius \( r = |k - 14| \).
4. Step 4: Set up the system of equations: Since the radius is both \( k \) and \( |k - 14| \), we equate these two expressions: \[ k = |k - 14| \] Solving this gives \( k = 7 \).
5. Step 5: Find the center and radius: The center of the circle is \( (9, 7) \), and the radius is \( 7 \).
6. Step 6: Write the equation of the circle: The equation of the circle is: \[ (x - 9)^2 + (y - 7)^2 = 7^2 \]
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