The values 'h' and 'k' represent the center of a circle when the circle's equation is expressed in standard form. Here's how to find them:
Understanding the Standard Equation of a Circle
The standard equation of a circle is:
(x – h)² + (y – k)² = r²
Where:
- (h, k) is the center of the circle
- r is the radius of the circle
Method 1: Directly from the Standard Equation
If you are given the equation of a circle in standard form, identifying 'h' and 'k' is straightforward. Simply compare the given equation with the standard equation.
Example:
Consider the equation: (x – 4)² + (y + 3)² = 29
Comparing this with the standard equation (x – h)² + (y – k)² = r², we can deduce:
- h = 4 (Note the subtraction sign in the standard equation: x - h)
- k = -3 (Since y + 3 = y - (-3), k = -3)
- r² = 29, so r = √29
Therefore, the center of the circle is (4, -3).
Method 2: Completing the Square (When Given the General Equation)
Sometimes, the equation of a circle is given in the general form:
x² + y² + Dx + Ey + F = 0
In this case, you need to complete the square to convert it into the standard form.
Steps:
-
Rearrange the equation: Group the x terms and y terms together, and move the constant term to the right side of the equation:
(x² + Dx) + (y² + Ey) = -F
-
Complete the square for x: Take half of the coefficient of the x term (D/2), square it ((D/2)²), and add it to both sides of the equation.
(x² + Dx + (D/2)²) + (y² + Ey) = -F + (D/2)²
-
Complete the square for y: Take half of the coefficient of the y term (E/2), square it ((E/2)²), and add it to both sides of the equation.
(x² + Dx + (D/2)²) + (y² + Ey + (E/2)²) = -F + (D/2)² + (E/2)²
-
Rewrite as squared terms: Express the x and y terms as squared binomials.
(x + D/2)² + (y + E/2)² = -F + (D/2)² + (E/2)²
-
Identify h and k: Now the equation is in the standard form (x – h)² + (y – k)² = r². Therefore:
- h = -D/2
- k = -E/2
- r² = -F + (D/2)² + (E/2)²
Example:
Let's say you have x² + y² + 6x - 4y - 3 = 0
- Rearrange: (x² + 6x) + (y² - 4y) = 3
- Complete the square for x: (x² + 6x + 9) + (y² - 4y) = 3 + 9
- Complete the square for y: (x² + 6x + 9) + (y² - 4y + 4) = 3 + 9 + 4
- Rewrite: (x + 3)² + (y - 2)² = 16
- Identify: h = -3, k = 2, r = 4. Center: (-3, 2)
Summary
Finding the values of 'h' and 'k' (the coordinates of the center) of a circle is a key skill in coordinate geometry. Depending on whether you have the standard form or the general form of the circle's equation, you can either directly read the values or complete the square to convert to standard form and then identify 'h' and 'k'.
