Intersection of a Straight Line and Circle on Argand Plane

The Argand plane is a geometric representation of complex numbers where each number is represented as a point with coordinates corresponding to its real and imaginary parts. In this context, a circle and a straight line can be represented on the Argand plane, and their intersection points, if any, can be determined using algebraic methods.

Representation of a Circle on the Argand Plane

A circle on the Argand plane is defined by the equation:

$$ (z - z_0)(\overline{z} - \overline{z_0}) = r^2 $$

where:

• $z$ is a complex number representing a point on the circle,
• $z_0$ is the center of the circle (also a complex number),
• $\overline{z}$ and $\overline{z_0}$ are the complex conjugates of $z$ and $z_0$ respectively,
• $r$ is the radius of the circle.

In terms of real and imaginary parts, if $z = x + yi$ and $z_0 = a + bi$, the equation becomes:

$$ ((x - a) + (y - b)i)((x - a) - (y - b)i) = r^2 $$ $$ (x - a)^2 + (y - b)^2 = r^2 $$

Representation of a Straight Line on the Argand Plane

A straight line on the Argand plane can be represented by the equation:

$$ Re(\overline{w}z) + c = 0 $$

where:

• $w$ is a complex number representing the direction of the line,
• $z$ is a complex number representing a point on the line,
• $c$ is a real number representing the line's offset from the origin.

In terms of real and imaginary parts, if $w = u + vi$ and $z = x + yi$, the equation becomes:

$$ u(x) + v(y) + c = 0 $$

Intersection Points

To find the intersection points of a straight line and a circle on the Argand plane, we need to solve the system of equations given by the circle and the line simultaneously.

Steps to Find Intersection Points

1. Write down the equations of the circle and the line.
2. Substitute the expression for $x$ or $y$ from the line equation into the circle equation.
3. Solve the resulting quadratic equation for the remaining variable.
4. Substitute the solutions back into the line equation to find the corresponding values for the other variable.
5. Write down the complex numbers corresponding to the intersection points.

Example

Let's find the intersection points of the circle with center $z_0 = 2 + 3i$ and radius $r = 4$, and the line with direction $w = 1 - i$ and offset $c = -5$.

Circle Equation

$$ (z - (2 + 3i))(\overline{z} - (2 - 3i)) = 4^2 $$ $$ (z - 2 - 3i)(\overline{z} - 2 + 3i) = 16 $$

Line Equation

$$ Re((1 + i)\overline{z}) - 5 = 0 $$ $$ (1 + i)(x - yi) - 5 = 0 $$ $$ (x + y) + (x - y)i - 5 = 0 $$ $$ x + y = 5 $$

Solving for Intersection Points

Substitute $x = 5 - y$ into the circle equation:

$$ ((5 - y) - 2 - 3i)((5 - y) - 2 + 3i) = 16 $$ $$ (3 - y - 3i)(3 - y + 3i) = 16 $$ $$ (3 - y)^2 + 9 = 16 $$ $$ y^2 - 6y + 9 = 7 $$ $$ y^2 - 6y + 2 = 0 $$

Solving the quadratic equation for $y$ gives us two possible values for $y$. Let's call them $y_1$ and $y_2$. For each of these, we can find the corresponding $x$ values by substituting back into $x + y = 5$.

Finally, the intersection points on the Argand plane are the complex numbers $z_1 = x_1 + y_1i$ and $z_2 = x_2 + y_2i$.

Summary Table

Feature	Circle	Straight Line
Equation	$(z - z_0)(\overline{z} - \overline{z_0}) = r^2$	$Re(\overline{w}z) + c = 0$
Parameters	Center $z_0$, Radius $r$	Direction $w$, Offset $c$
Real and Imaginary Parts	$(x - a)^2 + (y - b)^2 = r^2$	$u(x) + v(y) + c = 0$
Intersection Points	Solve system of equations	Solve system of equations

Conclusion

Finding the intersection points of a straight line and a circle on the Argand plane involves solving a system of equations that represent the geometric figures. By following the steps outlined above and using algebraic methods, one can determine the exact points of intersection, if they exist. This process is essential in complex analysis and has applications in various fields, including engineering and physics.