Statement-1: If z is a complex number such that |(2 z+3)/(3 z+1)|=√((5)/(3)), then z lies on a c

step 1 Hi, we're given here. Statement here, statement one, this equation. Let us just check that the l

Statement-1: If z is a complex number such that |(2 z+3)/(3...

Key Concepts

Complex Modulus

The modulus of a complex number, representing its distance from the origin in the complex plane, provides a bridge between algebra and geometry. It is often used to translate complex equations into geometric loci, such as circles or lines, by characterizing the distance relationships between points in the complex plane.

Circle Representation in the Complex Plane

Many geometric shapes, notably circles, can be represented by equations involving the modulus of complex numbers. Equations of the form |z - c| = r describe a circle with center c and radius r, and more generally, modulus equations involving ratios translate into circles or lines, depending on the parameters involved.

Apollonius Circle

An Apollonius circle is defined as the set of points in the complex plane for which the ratio of distances to two fixed points remains constant. This concept underlies expressions of the form |(z - z?)/(z - z?)| = k (with k ? 1), which represent a circle. It generalizes the idea of equally distant points (k = 1, yielding a straight line) to circles when the ratio is different from one.

Möbius Transformation

A Möbius transformation, or fractional linear transformation, is a function of the form (az + b)/(cz + d) that maps the complex plane to itself. An important property of these transformations is that they map circles and lines to circles and lines, preserving the general geometric structure of the set described by modulus equations, which is why they are central to solving problems linking algebraic expressions and geometric loci.

Transcript

00:01 Hi, we're given here.

00:04 Statement here, statement one, this equation.

00:08 Let us just check that the lighten of circle for this equation.

00:13 Statement to be given here, ab belongs to complex numbers, k even to real values, k not equal to 1, k2 0, and we have this equation here.

00:22 So we're going to check here, these two statements.

00:24 So let's first work on the second treatment here we have.

00:27 So if you just work on it, so i'll just give an ass here.

00:37 Here, question is for statement here.

00:40 We get mod z plus a equal k times more z plus b, or the square both sides.

00:49 And we know that complex number, multiply with this conjugate.

00:52 It gives more square.

00:54 So we get z plus a times the conjugate of z plus a, they conjugate plus a conjugate plus a conjugate.

01:01 That equals we have the next part is k square that's constant.

01:06 Next we have more z plus b whole square that's d plus b times.

01:10 V conjugate plus b conjugate that's simplify that so we get from here z times the conjugate plus a conjugate z plus a z conjugate positive a mod square equals we get a square times z conjugate next we have b conjugate z plus b z conjugate plus b mok squared.

01:42 Now we'll simplify this.

01:43 So we get it as one negative k squared z, z, we can't, plus we will get a conjugate negative k square b conjugate z.

02:02 Next we get plus a, plus a negative, k square b, z conjugate.

02:11 And the last journalist, mod a square negative k square, mod, b squared.

02:17 That equals a.

02:19 The next step is we just, here we have 180 case with a real value, a constant.

02:24 We just divide all the equation by that.

02:25 So we get the, the conjugate, positive, a conjugate negative k square, b conjugate over one negative k square, v, positive a negative, k square b over one negative k square times the conjugate, positive mod a square, negative k square, mod b square, or we have one negative t square.

02:54 That's going to be equal to zero.

02:56 This equation, we got now.

02:57 I just compared here, this the general equation of the circle that is given as z, z, z, the d not, plus we have a conjugate z plus a, z conjugate plus c, and a real value, that is equal to 0.

03:15 So it's match here.

03:16 It is matching with the equation of a circle.

03:21 Here we have.

03:22 This is the real value here we have.

03:24 And this is a conjugate, this is a here...

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