4BCircles

Circles in the Complex Plane ( Ellipses no longer in course)

Circles
 * {z: |z| = r}** forms a circle with a radius of r, centered at the origin.


 * {This rule should be compared to the geometric definition of a circle }


 * {The rule states that z is the collection of points where the distance to the origin is constant, r}


 * {z: |z** – **(a + bi)| = r}**
 * translates the circle to a centre at (a + bi)


 * {This version of the rule states that z is the collection of points where the distance to the point (a + bi) is constant and equal to r}

Substituting **z = x + yi** into this will give the cartesian equation of a circle

math \\ . \qquad |(x + yi) - (a + bi)| = r \\. \\ . \qquad |(x - a) + (y - b)i| = r math

math \\ . \qquad \sqrt {(x-a)^2 + (y-b)^2} = r \\. \\ . \qquad (x-a)^2 + (y-b)^2 = r^2 math

ELLIPSES AND HYPERBOLAS NO LONGER IN COURSE: Ellipses

In the cartesian plane, the equation of an ellipse is: math . \qquad \dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1 math

This has a center at (h, k), an x-radius of a and a y-radius of b.

The equivalent shape can be produced in the complex plane by substituting **x = Re(z)** and **y = Im(z)** math . \qquad \dfrac{ \big( Re(z)-h \big)^2}{a^2} + \dfrac{ \big( Im(z)-k \big)^2}{b^2} = 1 math

The geometric definition of an ellipse is based on two focii, F 1 and F 2.
 * The ellipse is defined as the set of points P
 * where the distance F 1 P __**plus**__ the distance F 2 P equals a constant 2a,
 * where 2a is the length of the major axis
 * (a = x-radius if major axis is horizontal).

... ... |F 1 P| + |F 2 P| = 2a

Compare this to the complex equation for an ellipse: math . \qquad |z+c|+|z-c|=2a math


 * Provided a > |c|**, this produces an __ellipse__ in the complex plane,
 * centred at the origin,
 * with focii at (+c, 0) and (–c, 0).

This can be confirmed by substituting **z = x + yi** and simplifying:

Example (see text, Eg8, p170)

Find the cartesian equation and hence show that |z – 2| + |z + 2| = 8 is an ellipse.

__**Solution:**__

math \\ . \qquad |(x-2) + yi| + |(x+2) + yi| = 8 \\. \\ . \qquad \sqrt{(x-2)^2+y^2} + \sqrt{(x+2)^2 + y^2} = 8 math

subtract one square root from both sides

math . \qquad \sqrt{(x-2)^2+y^2} = 8 - \sqrt{(x+2)^2 + y^2} math

square both sides using (a – b)² = a² – 2ab + b².

math . \qquad (x-2)^2+y^2 = 64 - 16\sqrt{(x+2)^2 + y^2} + (x+2)^2 + y^2 math

Expand and Simplify math \\ . \qquad x^2-4x+4+y^2 = 64 - 16\sqrt{x^2+4x+4 + y^2} + x^2+4x+4 + y^2 \\. \\ . \qquad -8x-64 = - 16\sqrt{x^2+4x+4 + y^2} \\. \\ . \qquad x+8 = 2\sqrt{x^2+4x+4 + y^2} math

Square both sides math \\ . \qquad x^2+16x+64 = 4(x^2+4x+4 + y^2) \\ .\\ . \qquad x^2+16x+64 = 4x^2+16x+16+4y^2 \\. \\ . \qquad 48 = 3x^2 + 4y^2 math

Divide both sides by 48 and simplify math \\ . \qquad \dfrac {3x^2}{48} + \dfrac {4y^2}{48} = 1 \\. \\ . \qquad \dfrac {x^2}{16} + \dfrac {y^2}{12} = 1 math

This is an ellipse, centre at (0, 0)

math . \qquad a = 4 \\. \\ . \qquad b=\sqrt{12} math

Hyperbola

In the cartesian plane, the equation of a hyperbola is: math . \qquad \dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1 math


 * This has a center at (h, k),
 * with vertices at
 * (h – a, k) and
 * (h + a, k)

math . \qquad \text{and equations of asymptotes: } \\. \\ . \qquad y-k = \pm \dfrac{b}{a}\big( x-h \big) math

The equivalent shape can be produced in the complex plane by substituting **x = Re(z)** and **y = Im(z)** math . \qquad \dfrac{ \big( Re(z)-h \big)^2}{a^2} - \dfrac{ \big( Im(z)-k \big)^2}{b^2} = 1 math

The geometric definition of a hyperbola is based on two focii, F 1 and F 2.
 * The hyperbola is defined as the set of points P
 * where the distance F 1 P __minus__ the distance F 2 P equals a constant 2a,
 * where 2a is the distance between the two vertices.
 * F 1 P – F 2 P = 2a

Compare this to the complex equation for a hyperbola: math . \qquad |z+c|-|z-c|=2a math


 * Provided a < |c|**,
 * this produces a __hyperbola__ in the complex plane,
 * centred at the origin, with focii at (+c, 0) and (–c, 0).

This can be confirmed by substituting **z = x + yi** and simplifying.

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