Choose a representation
Complete each variable square separately and balance added constants. For an ellipse compare denominator sizes. For a horizontal hyperbola, asymptote slopes are plus or minus b/a.
Worked example: follow the quantity
The circle x²+y²−6 x+10 y=10 has center (h,k). What is h? Complete the x square, using half the x coefficient. Read the center from (x−h)², reversing the sign inside. Substitute the given numbers: 6 ÷ 2 = 3. Therefore the requested value is 3.
Contrast a plausible mistake
For that example, the tempting answer 9 follows this error: Report the square-completion constant instead of the center. The requested quantity is 3. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
An ellipse is given by x²/9+y²/25=1, with 5>3. What is the length of its major axis? The larger denominator is the square of the major semiaxis. Take its square root and double to obtain the full axis. Substitute the given numbers: 2 × 5 = 10. Therefore the requested value is 10.
Transfer before checking
The circle x²+y²−8 x+12 y=10 has center (h,k). What is h? Try without viewing the answer. Check: 4. Complete the x square, using half the x coefficient. Read the center from (x−h)², reversing the sign inside. Substitute the given numbers: 8 ÷ 2 = 4.
Scope and conditions
A radius is the square root of the radius squared. An ellipse full axis is twice its semiaxis. Hyperbola denominators do not use the same ratio as their asymptote slopes. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
The circle x²+y²−6 x+10 y=10 has center (h,k). What is h?
- Complete the x square, using half the x coefficient.
- Read the center from (x−h)², reversing the sign inside.
- Substitute the given numbers: 6 ÷ 2 = 3.
- Therefore the requested value is 3.
Compare the other answer choices
Report the square-completion constant instead of the center.
Use the sign inside the squared binomial as the center coordinate.
Report the center y-coordinate instead.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content