Let \(S={z∈C:|z−2|≤1,and(1+i)+\overline{z}(1−i)≤2}. \)Let \(|z−4i|\) attains minimum and maximum values, respectively, at z1∈s and z2∈s. If \(5(|z_1|^2+|z_2|^2=α+β√5\) where α and β are integers, then the value of α+β is equal to _________.
Correct Answer: 26
Solution and Explanation
The correct answer is: 26
S represents the shaded region shown in the diagram.
Clearly, z1 will be the point of intersection of PA and given circle.
PA : 2x + y = 4 and given circle has equation
(x – 2)2 + y2 = 1.
On solving, we get
\(z_1=()+\frac{2}{5}i=|z_1|^2=5-\frac{4}{\sqrt5}\)
z2 will be either B or C.
So
\(5(|z_1|^2+|z_2|^2=30-4√5\)
Clearly α = 30 and β = –4 ⇒α + β = 26
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Concepts Used:
Relations
A relation in mathematics defines the relationship between two different sets of information. If two sets are considered, the relation between them will be established if there is a connection between the elements of two or more non-empty sets. Therefore, we can say, ‘A set of ordered pairs is defined as a relation.’
Read Also: Relation and Function
Types of Relations:
There are 8 main types of relations which are:
- Empty Relation - An empty relation is one in which there is no relation between any elements of a set.
- Universal Relation - A universal is a type of relation in which every element of a set is related to each other. Now one of the universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation, R = A × A
- Identity Relation - In an identity relation, every element of a set is related to itself only. For example, in a set A = {a, b, c}, the identity relation will be I = {a, a}, {b, b}, {c, c}.
- Inverse Relation - It is seen when a set has elements which are inverse pairs of another set. For example if set A = {(a, b), (c, d)}, then inverse relation will be R-1 = {(b, a), (d, c)}.
- Reflexive Relation - If every element of set A maps for itself, then set A is known as a reflexive relation.It is represented as a∈ A, (a,a) ∈ R.
- Symmetric Relation - A relation R on a set A is known as asymmetric relation if (a, b) ∈R then (b, a) ∈R , such that for all a and b ∈A.
- Transitive Relation - For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation, aRb and bRc ⇒ aRc ∀ a, b, c ∈ A
- Equivalence Relation - If a relation is reflexive, symmetric and transitive at the same time it is known as an equivalence relation.
Representation of Relations:
There are two ways by which a relation can be represented-
- Roster method
- Set-builder method
The roster form and set-builder for for a set integers lying between -2 and 3 will be-
Roster form
I= {-1,0,1,2}
Set-builder form
I= {x:x∈I,-2<x<3}