Let $z \in C$ be such that $\frac{z^2 +3i}{z-2+i} = 2+3i$. Then the sum of all possible values of $z^2$ is
$-19-2i$
We are given a complex equation $\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$ and we need to find the sum of all possible values of $z^2$. This involves solving a quadratic equation in $z$ and then calculating the sum of the squares of its roots.
Step 1: Simplify the Equation
First, we clear the denominator by multiplying both sides by $(z - 2 + i)$:
$z^2 + 3i = (2 + 3i)(z - 2 + i)$
$z^2 + 3i = (2 + 3i)z - (2 + 3i)(2 - i)$
Now, let's compute the product $(2 + 3i)(2 - i)$:
$(2 + 3i)(2 - i) = 4 - 2i + 6i - 3i^2$
Since $i^2 = -1$, we have:
$4 + 4i + 3 = 7 + 4i$
Substitute this back into the equation:
$z^2 + 3i = (2 + 3i)z - (7 + 4i)$
Step 2: Form the Quadratic Equation
Rearrange all terms to one side to form a standard quadratic equation $az^2 + bz + c = 0$:
$z^2 - (2 + 3i)z + (7 + 4i + 3i) = 0$
$z^2 - (2 + 3i)z + (7 + 7i) = 0$
Step 3: Calculate the Sum of the Squares of the Roots
Let the roots of the quadratic equation be $z_1$ and $z_2$. These are the possible values of $z$. We want to find the sum of all possible values of $z^2$, which is $z_1^2 + z_2^2$.
From Vieta's formulas, we know:
- Sum of roots: $z_1 + z_2 = \frac{-b}{a} = 2 + 3i$
- Product of roots: $z_1 z_2 = \frac{c}{a} = 7 + 7i$
Using the algebraic identity $z_1^2 + z_2^2 = (z_1 + z_2)^2 - 2z_1 z_2$:
$z_1^2 + z_2^2 = (2 + 3i)^2 - 2(7 + 7i)$
Expand $(2 + 3i)^2$:
$(2 + 3i)^2 = 4 + 12i + 9i^2 = 4 + 12i - 9 = -5 + 12i$
Expand $2(7 + 7i)$:
$2(7 + 7i) = 14 + 14i$
Subtract the two results:
$z_1^2 + z_2^2 = (-5 + 12i) - (14 + 14i)$
$z_1^2 + z_2^2 = -5 - 14 + 12i - 14i = -19 - 2i$
Final Answer
The sum of all possible values of $z^2$ is $-19 - 2i$.
Similar Questions
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
- The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is
- Let $f$ be a function such that $3f(x) + 2f\left(\frac{m}{19x}\right) = 5x, x \neq 0$, where $m = \sum_{i=1}^{9} (i)^2$. Then $f(5) - f(2)$ is equal to
- The smallest positive integral value of $a$, for which all the roots of $x^4 - ax^2 + 9 = 0$ are real and distinct, is equal to
- $\left(\frac{1}{3} + \frac{4}{7}\right) + \left(\frac{1}{3^2} + \frac{1}{3} \times \frac{4}{7} + \frac{4^2}{7^2}\right) + \left(\frac{1}{3^3} + \frac{1}{3^2} \times \frac{4}{7} + \frac{1}{3} \times \frac{4^2}{7^2} + \frac{4^3}{7^3}\right) + \dots$ upto infinite terms, is equal to
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
Important Questions from Algebra
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
