Problems in g, f and c Revision Notes for Leaving Cert Mathematics

Problems in g, f and c

What Are $g, f,$ and $c$ in a Circle Equation?

In the general equation of a circle:

x^2 + y^2 + 2gx + 2fy + c = 0

$g, f,$ and $c$ are coefficients that determine the circle's centre and radius.
Centre: $(-g, -f)$
Radius: $r = \sqrt{g^2 + f^2 - c}$

Types of Problems Involving $g, f,$ and $c$ :

Finding the Centre and Radius:

Given the equation of the circle, extract $g, f,$ and $c$ to compute the centre and radius.

Verifying Points on a Circle:

Check if a point $(x, y)$ satisfies the equation.

Rewriting in Standard Form:

Convert the general form to:

(x - h)^2 + (y - k)^2 = r^2

Special Cases:

If $g^2 + f^2 = c$ , the radius is zero, and the circle reduces to a point.
If $g^2 + f^2 - c < 0$ , the equation does not represent a real circle.

Worked Examples

infoNote

Example 1: Find the Centre and Radius

Problem: For the circle $x^2 + y^2 - 4x + 6y - 12 = 0$ , find the centre and radius.

Solution:

Step 1: Identify $g, f, c$ :

2g = -4 \implies g = -2, \quad 2f = 6 \implies f = 3, \quad c = -12

Step 2: Calculate the centre:

(-g, -f) = (2, -3)

Step 3: Calculate the radius:

r = \sqrt{g^2 + f^2 - c} = \sqrt{(-2)^2 + 3^2 - (-12)}

= \sqrt{4 + 9 + 12} = \sqrt{25} = 5

Answer: The centre is $(2, -3)$ , and the radius is $5$

infoNote

Example 2: Rewrite in Standard Form

Problem: Rewrite $x^2 + y^2 + 8x - 10y + 20 = 0$ in standard form.

Solution:

Step 1: Group $x$ and $y$ terms:

(x^2 + 8x) + (y^2 - 10y) = -20

Step 2: Complete the square for $x$ :

x^2 + 8x \to (x + 4)^2 - 16

Step 3: Complete the square for $y$ :

y^2 - 10y \to (y - 5)^2 - 25

Step 4: Substitute back:

(x + 4)^2 - 16 + (y - 5)^2 - 25 = -20

(x + 4)^2 + (y - 5)^2 = 21

Answer: The equation in standard form is $(x + 4)^2 + (y - 5)^2 = 21$

Summary

The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$
Centre: $(-g, -f)$
Radius: $r = \sqrt{g^2 + f^2 - c}$
Key problem types:
- Finding centre and radius.
- Checking if points lie on the circle.
- Converting to standard form.
Ensure to check conditions $g^2 + f^2 - c > 0$ for real circles.

Problems in g, f and c (Leaving Cert Mathematics): Revision Notes