Complex Numbers and Circle Geometry
Complex Numbers
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A complex number is any number that can be written in the form a + bi where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i² = -1.
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The real part of a complex number a + bi is a, and the imaginary part is b.
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Complex numbers can be added, subtracted, multiplied, and divided much like real numbers, but with the added step of simplifying terms containing i².
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The conjugate of a complex number a + bi is the number a - bi. Conjugates are used to find the real part of fractions involving complex numbers and the modulus of a complex number.
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The modulus or absolute value of a complex number, denoted by |z|, is given by √(a² + b²), where a and b are the real and imaginary parts of z.
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The argument of a complex number z = a + bi, denoted by arg(z), is the angle between the positive real axis and the line segment joining the origin (0,0) to the point representing z on the complex plane. It is found using the formula arg(z) = tan⁻¹(b/a).
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The exponential form of a complex number is z = re^(iθ), where r is the modulus of z and θ is the argument of z. It is useful when multiplying and dividing complex numbers.
Circle Geometry
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A circle is the locus of all points in a plane that are equidistant from a fixed point called the centre.
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The equation of a circle with centre at (h, k) and radius r is given by (x - h)² + (y - k)² = r². If the centre of the circle is at the origin (0,0), the equation simplifies to x² + y² = r².
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Tangents to a circle at point P are lines that touch the circle at P but do not intersect the circle anywhere else. The tangent is perpendicular to the radius at the point of contact.
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The chord of a circle is a line segment which joins any two points on the circle.
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A line segment, drawn from the centre of the circle to the midpoint of a chord, bisects the chord and is perpendicular to it.
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If PQ is a chord of a circle and A is the point on the chord such that AP: AQ = m : n (m ≠ n), then OP² = r² - mn/r(m + n)AQ². This is often referred to as the Intercept Theorem.
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When a chord of a circle and a tangent at one end of the chord are extended to meet at a point outside the circle, the alternate segment theorem states that the angle between the tangent and the chord is equal to the angle in the alternate segment (formed by the chord).