Locus of the complex number z satisfying Column I Column II (a) |z-1|=|z+i| is (p) Circle (b) |z

Locus of the complex number z satisfying Column I Column II...

Key Concepts

Complex Plane Representation

The complex plane represents complex numbers as points with horizontal (real) and vertical (imaginary) coordinates. This geometric perspective allows algebraic relationships between complex numbers to be visualized as curves or regions in a plane.

Modulus as a Distance Measure

The modulus of a complex number, expressed as |z|, measures the distance of the point from the origin. When written in the form |z - a|, it represents the distance between the point z and a fixed point a in the plane, translating algebraic conditions into geometric distances.

Geometric Locus

A locus is the set of all points that satisfy a given condition or equation. In the context of complex numbers, algebraic equations involving z define specific geometric shapes such as circles, ellipses, or lines in the complex plane.

Circle

In the complex plane, a circle is defined as the set of points that maintain a constant distance (the radius) from a fixed center. Equations of the form |z - a| = r directly correspond to this definition.

Ellipse

An ellipse is defined as the set of points for which the sum of the distances from two fixed foci is constant. This property allows an algebraic equation that sets a constant sum of distances to be recognized as the equation of an ellipse.

Straight Line

A straight line in the complex plane is a one-dimensional set of points that extend infinitely in two opposite directions. Loci defined by conditions such as equidistance between two points can often result in linear equations representing straight lines.

Apollonius Circle

An Apollonius circle is the set of points in the complex plane that maintain a constant ratio of distances to two fixed points. This concept generalizes the idea of equidistance (which gives a perpendicular bisector when the ratio is 1) to produce a circle when the ratio differs from 1.

Complex Transformations

Complex transformations, including rotations and translations, arise when complex numbers are multiplied or added by other complex numbers. These operations can change the orientation or position of geometric figures in the complex plane, helping to interpret and simplify locus equations.

Transcript

00:01 Let us solve this match the following problem based on complex numbers.

00:06 Let's see the first part a so we need to match a with one of these pqrs so mod z minus 1 so what do you mean by geometrically mod z minus z not in complex numbers the geometrically mod z minus z not mean the distance between z and z not the distance between the points is z minus z not so what do you mean by mod z minus 1? so what is z minus 1? so what z minus 1 is basically? distance between z i so this distance between z and 1 comma 0 because 1 1 means 1 comma 0 because in complex numbers a plus i b means a comma b in the coordinate plane two -dimensional plane and similarly what do you mean by z plus i it is the distance between 0 minus 1 so it is 1 comma 0 minus 1 now it says that the distance between the locus of a point which is moving in such way that the distance between 1 comma 0 is same as the distance between 0 .1.

01:10 So let's plot it geometrically.

01:13 It looks like this.

01:13 So we have 1 comma 0 here, 0 comma minus 1 here.

01:19 So now if i just join this, this is actually a straight line.

01:25 Now what do you think the distance, i mean the locus of a point which is moving such way that it is maintaining the same distance between these two points first of all one of the candidate is the midpoint because the midpoint is actually equidistant from the both not only midpoint it will be a point here because i can form an isoseless triangle so what do you mean by isoseless triangle this distance is same as this distance not only that i can form another isoseless triangle so where this point is another candidate where this distance is same as this distance so all these points form a locus now if we join all this including this midpoint it looks something like this so that and that will be perpendicular bisector because it's a midpoint and it should be a perpendicular bisector why this perpendicular is this simple property of isoseless triangle so if you take an isoseless triangle if you take this particular vertex if you join the midpoint of the opposite side with the congriency axioms you will see that this angle is same as this angle but if these two angles are same and since they add up to 180 degrees each should be what 90 degrees okay so the locus of a point which moves in such way that its distance from one point says z1 is same as its distance from another point z2 is actually a perpendicular bisector it's a perpendicular bisector of the points z1 and z2 and since the perpendicular bisector is obviously a straight line the option for a matching with s so a matches with s now let's see this part b it's mod z minus 4i plus mod z plus 4i is equal to 10 that means the sum of the distances from 0 comma 4 because 4i 0 comma 4 and plus 4 i means the sum of the distance is actually 10 that means a point is moving in such a way that it's maintaining the sum of the distances from these two fixed points as 10 and this resembles some something like a property of ellipse ellipse because in ellipse what happens is this is an ellipse so this is focus one this is another focus two so basically we call it as focai the point p if you take any randomly any point p on the ellipse it always satisfies this distance plus this distance is actually a constant everywhere irrespective of the position of p on the ellipse right now let's verify it i'm not saying that it should be ellips let's verify when does it becomes ellips is this is ae comma zero and this is minus sorry this is minus ae comma zero and this is is ae comma zero the distance between these two points is two ae so that means this is my foci s dash s s s and this is s dash so that means when you measure the distance between these two it's eight so that means eight is same as two ae so that means two ae is equal to eight so, that means e is equal to 8 by 2a, so that is nothing but 4 by a.

04:41 And the beautiful property of ellipse is when you add this distance and this distance, it will be a constant and that constant is actually the length of major axis of the ellipse.

04:50 And we know that this is negative a comma 0 and this is positive a comma 0.

04:54 And i'm talking about the standard ellipse x square by a square plus y square by b square equal 1.

04:58 In fact, it's true for any ellipse, in fact.

05:00 But this is the simplest case i'm taking.

05:01 So for better understanding.

05:03 So it's minus a comma 0 to a comma 0 the distance is 2a.

05:07 So some of the distances is actually 2a.

05:10 Now let's measure the sum of the distances.

05:14 I mean, let's measure the value of 2a.

05:16 So 2a is actually this because it's given 10.

05:18 So 2a is equal to 10.

05:20 So that means a is equal to 5.