Solved Problems: Understanding (x-h)^2 + (y-k)^2 = r^2 examples

📚 Quick Study Guide

📐 The equation $(x-h)^2 + (y-k)^2 = r^2$ represents a circle in the Cartesian plane. 📍 $(h, k)$ represents the center of the circle. 📏 $r$ represents the radius of the circle. ➕ To find the equation, substitute the known values of the center $(h, k)$ and the radius $r$ into the formula. ➖ If the center is at the origin $(0,0)$, the equation simplifies to $x^2 + y^2 = r^2$. 💡 Completing the square might be necessary to transform a given equation into the standard form. 📝 Remember that $r^2$ is the square of the radius, so make sure to square the radius when writing the equation!

🧪 Practice Quiz

1. What is the center of the circle represented by the equation $(x-3)^2 + (y+2)^2 = 16$?
1. (3, 2)
2. (-3, -2)
3. (3, -2)
4. (-3, 2)


2. What is the radius of the circle represented by the equation $(x+1)^2 + (y-4)^2 = 25$?
1. 5
2. 25
3. 625
4. 12.5


3. The center of a circle is at (0, 0) and its radius is 7. What is the equation of the circle?
1. $x^2 + y^2 = 7$
2. $x^2 + y^2 = 14$
3. $x^2 + y^2 = 49$
4. $x^2 + y^2 = \sqrt{7}$


4. What is the equation of a circle with center at (-2, 5) and radius 3?
1. $(x-2)^2 + (y+5)^2 = 9$
2. $(x+2)^2 + (y-5)^2 = 9$
3. $(x-2)^2 + (y+5)^2 = 3$
4. $(x+2)^2 + (y-5)^2 = 3$


5. Which of the following points lies on the circle $(x-1)^2 + (y+1)^2 = 4$?
1. (1, 1)
2. (3, -1)
3. (1, -1)
4. (0, 0)


6. The equation of a circle is $(x+4)^2 + (y+3)^2 = 1$. What is the diameter of the circle?
1. 1
2. 2
3. 4
4. 0.5


7. What is the center of the circle given by the equation $x^2 + y^2 - 4x + 6y - 3 = 0$? (Hint: Complete the square)
1. (2, -3)
2. (-2, 3)
3. (4, -6)
4. (-4, 6)