Question

Determine the equation of the circle passing through (-4,-2).
1. (1,-1) lies in the circle.
2. The center of the circle is the origin.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

Remember that it takes three non-collinear points to uniquely define a circle. Statement (1) only provides a total of two points, which is not enough. Statement (2) provides the center, which is a very powerful piece of information. Once you have the center, you only need one point on the circumference to find the radius and complete the equation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To determine the unique equation of a circle, we need to know its center (\(h, k\)) and its radius (\(r\)). The standard equation of a circle is \( (x-h)^2 + (y-k)^2 = r^2 \). We are given that one point, (-4, -2), is on the circle.
Step 2: Detailed Explanation:
Analyzing Statement (1): (1,-1) lies in the circle.
Assuming "lies in the circle" means "lies on the circumference", this gives us a second point on the circle. So we know two points on the circle are P1(-4, -2) and P2(1, -1).
Let the center be C(h, k). The distance from the center to any point on the circle is the radius. Therefore, CP1 = CP2.
\[ \sqrt{(-4-h)^2 + (-2-k)^2} = \sqrt{(1-h)^2 + (-1-k)^2} \] Squaring both sides:
\[ (-4-h)^2 + (-2-k)^2 = (1-h)^2 + (-1-k)^2 \] \[ 16 + 8h + h^2 + 4 + 4k + k^2 = 1 - 2h + h^2 + 1 + 2k + k^2 \] \[ 20 + 8h + 4k = 2 - 2h + 2k \] \[ 18 + 10h + 2k = 0 \] \[ 5h + k = -9 \] This is a single linear equation with two variables (\(h, k\)). There are infinitely many possible centers that satisfy this equation (all lying on the line \(k = -9 - 5h\)). Since we cannot find a unique center, we cannot find a unique equation for the circle. Thus, statement (1) is not sufficient.
Analyzing Statement (2): The center of the circle is the origin.
This statement directly gives us the center of the circle: \( (h, k) = (0, 0) \).
The equation of the circle is \( (x-0)^2 + (y-0)^2 = r^2 \), which simplifies to \( x^2 + y^2 = r^2 \).
We know the circle passes through the point (-4, -2). We can substitute these coordinates into the equation to find the radius \( r \).
\[ (-4)^2 + (-2)^2 = r^2 \] \[ 16 + 4 = r^2 \] \[ r^2 = 20 \] Now we have both the center and the radius squared, so we can write the unique equation of the circle:
\[ x^2 + y^2 = 20 \] Since we can determine the unique equation, statement (2) is sufficient.
Step 3: Final Answer:
Statement (2) alone is sufficient, but statement (1) alone is not.