Circles

Step 1: Understanding the Problem
The circle touches the y
axis at a distance of 4 units from the origin, which means the center of the circle is at a distance of 4 units from the y
axis. Therefore, the x
coordinate of the center is .
The circle cuts off an intercept of 6 units on the x
axis, which means the length of the chord along the x
axis is 6 units.
Step 2: Determining the Center and Radius
Let the center of the circle be . Since the circle touches the y
axis, .
The equation of the circle is:
The circle cuts off an intercept of 6 units on the x
axis, so the distance from the center to the x
axis is , and the radius can be found using the chord length formula:
Step 3: Solving for the Center and Radius
Since the circle touches the y-axis, the radius .
Substituting into the chord length equation:
Therefore, the center of the circle is or .
Step 4: Writing the Equation of the Circle
The equation of the circle with center is:
Expanding this equation:
Simplifying:
Similarly, for the center :
Expanding this equation:
Simplifying:
Step 5: Matching with the Options
The correct equations are:
These equations correspond to option (A) when considering the correct coefficients.

Final Answer: The correct equation of the circle is (A) .