Equation of a Circle Test Answer Key with Scoring – Geometry

Question 1 Write the Equation (2 points)

r² = (3√2)² = 18.

(x + 4)² + (y − 7)² = 18

Scoring: 1 point for r², 1 point for the signs of the center.

Question 2 Read the Equation (2 points)

Center (−7, 3). Radius √50 = 5√2.

Center (−7, 3); radius 5√2

Scoring: 1 point for the center, 1 point for the radius.

Question 3 Circle from a Diameter (3 points)

Center = (−3 + 92, −5 + 12) = (3, −2).

r² = (9 − 3)² + (1 + 2)² = 36 + 9 = 45.

(x − 3)² + (y + 2)² = 45

Scoring: 1 point for the center, 1 point for r², 1 point for the equation.

Question 4 Complete the Square (3 points)

(x² + 10x + 25) + (y² − 6y + 9) = 2 + 25 + 9 = 36.

(x + 5)² + (y − 3)² = 36.

Center (−5, 3), radius 6

Scoring: 1 point for each added constant, 1 point for the standard form and center and radius.

Question 5 Inside or Outside (3 points)

Equation: (x + 1)² + (y − 2)² = 25.

A(1, 4): 4 + 4 = 8, less than 25, inside.

B(4, 2): 25 + 0 = 25, on the circle.

C(−5, 6): 16 + 16 = 32, greater than 25, outside.

A inside, B on, C outside

Scoring: 1 point for the equation, 1 point for A and B, 1 point for C and correct reasoning.

Question 6 Radio Station Range (3 points)

(a) (x − 7)² + (y + 2)² = 81.

(b) (1 − 7)² + (4 + 2)² = 36 + 36 = 72, which is less than 81, so the school is in range.

(a) (x − 7)² + (y + 2)² = 81 (b) Yes

Scoring: 1 point for the equation, 1 point for the distance squared, 1 point for the conclusion.

Question 7 Tangent to Both Axes (2 points)

In the second quadrant x is negative and y is positive, so the center is (−9, 9).

(x + 9)² + (y − 9)² = 81

Scoring: 1 point for the center, 1 point for the equation.

Question 8 Read a Graph (2 points)

The center is (−3, −2). The rightmost point is (0, −2), 3 units from the center, so r = 3.

(x + 3)² + (y + 2)² = 9

Scoring: 1 point for the center, 1 point for the radius and equation.