Question

In the rectangular coordinate system, the circle with center P is tangent to both the x- and y-axes.
Column A: The x-coordinate of P
Column B: The y-coordinate of P

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

Visualize the geometry. A circle that fits perfectly into a corner, touching both walls (the axes), must have its center at an equal distance from both walls. This means its x and y coordinates must be the same.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks us to compare the x- and y-coordinates of the center of a circle. The key information is that the circle is "tangent" to both axes. Tangent means it touches the axis at exactly one point.
Step 2: Detailed Explanation:
Let the coordinates of the center P be \((x_p, y_p)\). Let the radius of the circle be \(r\).
\begin{itemize} \item The distance from the center of a circle to a line that is tangent to it is equal to the radius. \item The distance from the center P\((x_p, y_p)\) to the x-axis (the line \(y=0\)) is \(|y_p|\). Since the circle is tangent to the x-axis, this distance must be the radius: \(r = |y_p|\). \item The distance from the center P\((x_p, y_p)\) to the y-axis (the line \(x=0\)) is \(|x_p|\). Since the circle is tangent to the y-axis, this distance must also be the radius: \(r = |x_p|\). \item Therefore, we can conclude that \(|x_p| = |y_p| = r\). \item From the provided diagram, the circle is located in the first quadrant, where both x and y coordinates are positive. Thus, we can drop the absolute value signs: \(x_p = y_p\). \end{itemize} Step 3: Final Answer:
The x-coordinate of P is equal to the y-coordinate of P. Therefore, the two quantities are equal.