Questions: Conic Sections: Circles

Question 1

What are the center and radius of the circle (x − 4)² + (y + 3)² = 25?

• A. Center (4, 3), radius 25
• B. Center (−4, −3), radius 5
• C. Center (4, −3), radius 5
• D. Center (4, 3), radius 5

Answer: (C)

The standard form is (x − h)² + (y − k)² = r². Matching: (x − 4)² tells us h = 4. The y-term is (y + 3)² = (y − (−3))², so k = −3. The radius is √25 = 5, not 25. Option D is the most common mistake — reading the y-coordinate as +3 instead of −3 because the equation shows '+3' inside the parentheses. The rule: the center's coordinate is the value you *subtract from* the variable to get the expression inside the parentheses, so (y + 3) = (y − (−3)) means y-coordinate is −3.

Question 2

A circle passes through all points exactly 6 units from (−2, 5). Which equation represents this circle?

• A. (x + 2)² + (y − 5)² = 6
• B. (x − 2)² + (y + 5)² = 36
• C. (x + 2)² + (y − 5)² = 36
• D. (x − 2)² + (y + 5)² = 6

Answer: (C)

The center is (−2, 5), giving (x − (−2))² + (y − 5)² = (x + 2)² + (y − 5)². The radius is 6, so the right side is r² = 36, not 6. Option A uses the correct center but forgets to square the radius — the most common arithmetic error. Options B and D have the center signs flipped: center (−2, 5) requires (x + 2)² and (y − 5)², not (x − 2)² and (y + 5)².

Question 3

The equation (x − 3)² + (y + 2)² = 16 represents a circle with center (3, 2) and radius 16.

• T. True
• F. False

Answer: False

Two errors are packed into this statement. First, the center's y-coordinate is −2, not 2: (y + 2)² = (y − (−2))² means k = −2. Second, the radius is √16 = 4, not 16. The right side of the standard form equation is r², so you must take the square root to find r. These are the two most common circle equation errors: misreading the sign of the center and forgetting to take the square root of the right side.

Question 4

The standard form of a circle's equation, (x − h)² + (y − k)² = r², is derived directly from the distance formula by squaring both sides of the equation for all points at distance r from center (h, k).

• T. True
• F. False

Answer: True

The derivation is direct: start with the geometric definition (all points (x, y) at distance r from (h, k)), apply the distance formula (√((x−h)² + (y−k)²) = r), and square both sides to get (x−h)² + (y−k)² = r². This is not a formula to memorize in isolation — it is the algebraic encoding of the geometric definition. Understanding this origin makes sign errors less likely: you know the equation says 'the distance from (x, y) to (h, k) equals r,' which tells you exactly where h and k appear and why the right side is r² not r.

Question 5

Explain why the x-coordinate of the center in (x + 5)² + (y − 2)² = 36 is −5, not 5. Use the geometric definition of a circle in your explanation.

Model answer: A circle is the set of all points at distance r from the center (h, k). The distance formula gives √((x−h)² + (y−k)²) = r, and squaring both sides gives (x−h)² + (y−k)² = r². The term (x + 5)² must match the form (x − h)², so we rewrite it: (x + 5)² = (x − (−5))², revealing that h = −5. The center's x-coordinate is the value you subtract from x to get the expression — here, you subtract −5, so the center is at x = −5. Reading (x + 5)² as 'center at x = 5' confuses the sign because the standard form subtracts h, so a positive term inside means a negative center coordinate.

This sign confusion is responsible for most errors in circle problems. The key habit is to rewrite the term in standard form before reading the center: always express as (x − h)², never read h directly from the coefficient. If the equation shows (x + 5), rewrite it as (x − (−5)) before identifying h = −5. The same logic applies to y: (y − 2)² directly gives k = 2, while (y + 3)² = (y − (−3))² gives k = −3.