Column I Column II (a) Minimum value of |z-2+i|+|z-2 i+1| is (p) 6 (b) Minimum value of |z1-z2

Column I Column II (a) Minimum value of |z-2+i|+|z-2 i+1|...

Key Concepts

Triangle Inequality

The triangle inequality is a fundamental result in complex analysis and metric spaces which states that for any two complex numbers, the modulus of their sum is no greater than the sum of their moduli. This inequality is widely used in problems involving bounds on distances, helping to establish minimum or maximum possible values when combined with other geometric insights.

Circle Loci

A circle in the complex plane is defined by an equation of the form |z - a| = r, where a is the center of the circle and r is its radius. This concept is useful for understanding constraints that restrict a complex number to a circle, enabling analysis of distances and relationships between points constrained to circular loci.

Optimization in the Complex Plane

Optimization in the context of complex numbers often involves finding the maximum or minimum values of distance-related expressions, subject to given constraints such as being on a circle or an ellipse. This requires a synthesis of geometric interpretations, the triangle inequality, and algebraic manipulation to determine conditions under which these extreme values occur.

Modulus as Distance

The modulus of a complex number, denoted by |z|, represents the distance of the point z from the origin in the complex plane. When expressions take the form |z - a|, they describe the Euclidean distance between the point z and the fixed point a. This concept is fundamental in questions involving the minimization or maximization of distances between points.

Complex Plane Representation

This concept involves treating complex numbers as points in a two?dimensional plane, where the real part represents the horizontal coordinate and the imaginary part represents the vertical coordinate. This geometric interpretation allows one to use distance and angle concepts from coordinate geometry when manipulating and analyzing complex numbers.

Ellipse Definition in the Complex Plane

An ellipse can be defined as the set of all points in the plane for which the sum of the distances to two fixed points (foci) is constant. In the context of complex numbers, expressions that involve the sum of two moduli, such as |z - a| + |z - b| = constant, describe an ellipse, provided that the constant is greater than the distance between the foci.

Transcript

00:01 In this question, we need to match the elements of column 1 with the elements of column 2.

00:06 The part a is saying that minimum value of of modulus z minus 2 plus iota plus modulus z minus 2 iota plus 1 will be.

00:36 We know that modulus z1 minus z2 is less than equals to modulus z1 plus modulus z2.

00:54 We can say that let's say modulus z1 is equals to modulus z minus 2 plus iota and modulus z 2 is equals to modular z minus 2 iota plus 1 substitute these values in this equation let's say this was equation 1 so we get modulus z minus 2 plus iota plus modulus z minus 2 aorta plus 1 greater than equals to modulus minus 2 plus iota minus z minus 2 iota plus 1 when we further solve it we get minus 2 iota plus 2 iota minus 1 and this will be equals to minus 3 plus 3 aota and this will be equal to under root of minus 3 square plus 3 square is equals to 3 root 2.

02:39 Therefore the correct match for the first part is a and s.

03:01 Now let's come to the part b.

03:14 The part b is saying that minimum value of of modulus z1 minus z2 where modulus z1 plus 1 minus iota is equal to 13 and modulus z3 minus 3 minus 4 aota is equal to 2 we can observe that these two are the equations of circle therefore modulus z 1 plus 1 minus iota is equal to 13 is a circle and its center is minus 1 comma 1 and its radius is 3.

04:27 Similarly, modulus z3 minus 3 minus 4 aota is equals to 2 is the circle.

04:37 Its center is 3 .4 and radius is 2.

04:56 Based on this information we can draw these circles and the circle is shown below.

05:11 So these are the two circles.

05:15 Let's say this point is c1.

05:18 This is c2, this point is a and this point is b.

05:30 The coordinates of c1 are minus 1 .1 and the coordinates of c2 are 3 .4.

05:45 Therefore we can write minimum value of modulus z1 minus z2 is equal to ab and this will be equals to c1b minus c1c2 plus c2a.

06:22 From the figure we can see that c1b is equal to 13 and c2a is equal to 2...