Intersections of lines and circles

To find the intersection points between a circle and a straight line, substitute the equation of the straight line into the circle's equation and solve for \( x \) and \( y \). Let’s work through this step by step.

Step 1: Start with the Equations

The equation of a circle is typically given as:

\[ x^2 + y^2 = r^2 \]

where \( r \) is the radius.

The equation of a straight line is usually written as:

\[ y = mx + c \]

where \( m \) is the gradient and \( c \) is the \( y \)-intercept.

Step 2: Substitute the Line Equation into the Circle Equation

Replace \( y \) in the circle's equation with \( mx + c \). For example, consider:

\[ x^2 + y^2 = 25 \quad \text{and} \quad y = x + 3 \]

Substitute \( y = x + 3 \) into \( x^2 + y^2 = 25 \):

\[ x^2 + (x + 3)^2 = 25 \]

Step 3: Expand and Simplify

Expand the squared term and simplify the equation:

\[ x^2 + (x^2 + 6x + 9) = 25 \]

Combine like terms:

\[ 2x^2 + 6x + 9 = 25 \]

Simplify further:

\[ 2x^2 + 6x - 16 = 0 \]

Step 4: Solve the Quadratic Equation

Factorise the quadratic equation:

\[ 2x^2 + 6x - 16 = 0 \]

Factor out the common factor \( 2 \):

\[ 2(x^2 + 3x - 8) = 0 \]

Factorise \( x^2 + 3x - 8 \):

\[ (x + 4)(x - 2) = 0 \]

So the solutions are:

\[ x = -4 \quad \text{and} \quad x = 2 \]

Step 5: Find the Corresponding \( y \)-Values

Substitute these \( x \)-values back into the line equation \( y = x + 3 \):

For \( x = -4 \):

\[ y = -4 + 3 = -1 \]

For \( x = 2 \):

\[ y = 2 + 3 = 5 \]

The intersection points are:

\[ (-4, -1) \quad \text{and} \quad (2, 5) \]

Step 6: Verify the Solutions

Check that the points satisfy both the circle and line equations. For example:

For \( (-4, -1) \):

\[ x^2 + y^2 = (-4)^2 + (-1)^2 = 16 + 1 = 25 \]

For \( (2, 5) \):

\[ x^2 + y^2 = 2^2 + 5^2 = 4 + 25 = 25 \]

Both points satisfy the circle equation.

Summary

• Substitute the straight line equation into the circle equation.
• Expand and simplify to form a quadratic equation.
• Solve the quadratic equation to find \( x \)-values.
• Substitute \( x \)-values back into the line equation to find \( y \)-values.
• Verify the solutions to ensure they satisfy both equations.