Question

The equation of a circle with center \( (5, 4) \) and touching the \( Y \)-axis is:

• A. \( x^2 + y^2 - 10x - 8y - 16 = 0 \)
• B. \( x^2 + y^2 - 10x - 8y - 16 = 0 \)
• C. \( x^2 + y^2 + 10x + 8y + 16 = 0 \)
• D. \( x^2 + y^2 - 10x - 8y + 16 = 0 \)

Hint:

To derive the equation of a circle, substitute the center coordinates and radius into the general formula, then expand and simplify the expression.

Answer 1

The Correct Option is D

The standard form of a circle's equation is: \[ (x - h)^2 + (y - k)^2 = r^2, \] where:
\( (h, k) \) denotes the center of the circle, and
\( r \) represents the radius. Step 1: Identify given data. The center of the circle is \( (5, 4) \), so we have \( h = 5 \) and \( k = 4 \). Since the circle touches the \( Y \)-axis, the radius \( r \) is equal to the \( x \)-coordinate of the center, thus \( r = 5 \). Step 2: Substitute values into the equation. Using the center and radius, the equation of the circle is: \[ (x - 5)^2 + (y - 4)^2 = 5^2. \] Step 3: Simplify the equation. First, expand each binomial: \[ (x - 5)^2 = x^2 - 10x + 25, \quad (y - 4)^2 = y^2 - 8y + 16. \] Now, add the terms: \[ x^2 - 10x + 25 + y^2 - 8y + 16 = 25. \] Simplifying further: \[ x^2 + y^2 - 10x - 8y + 16 = 0. \] Step 4: Check the corresponding option. The resulting equation matches option \( \mathbf{(2)} \). Final Answer: The equation of the circle is: \[ \boxed{x^2 + y^2 - 10x - 8y + 16 = 0}. \]