Question

One of the values of $\sqrt{24-70i} + \sqrt{-24+70i}$ is

• A. $2+12i$
• B. $12-2i$
• C. $-12+2i$
• D. $-12-2i$

Hint:

A quick method to find the square root of $a+ib$ is to use the formula: $\sqrt{a+ib} = \pm \left( \sqrt{\frac{\sqrt{a^2+b^2}+a}{2}} + i \cdot \text{sgn}(b) \sqrt{\frac{\sqrt{a^2+b^2}-a}{2}} \right)$. For $24-70i$, $|z|=74, a=24, b=-70$. $\sqrt{24-70i} = \pm \left( \sqrt{\frac{74+24}{2}} - i \sqrt{\frac{74-24}{2}} \right) = \pm(\sqrt{49} - i\sqrt{25}) = \pm(7-5i)$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The problem requires finding the sum of the square roots of two complex numbers which are negatives of each other. A complex number has two square roots. The sum will have four possible values depending on which roots we choose. We need to find one of these values among the given options.
Step 2: Key Formula or Approach
Let $z = 24-70i$. We need to evaluate $\sqrt{z} + \sqrt{-z}$.
Note that $\sqrt{-z} = \sqrt{-1 \cdot z} = \pm i \sqrt{z}$.
So the expression becomes $\sqrt{z} \pm i \sqrt{z} = \sqrt{z}(1 \pm i)$.
We first find the square root of $z = 24-70i$. Let $\sqrt{24-70i} = x+iy$. Then $(x+iy)^2 = 24-70i$.
This gives two equations: $x^2-y^2=24$ and $2xy=-70$.
We can also find $x^2+y^2 = \sqrt{24^2+(-70)^2}$.
Step 3: Detailed Explanation
Let $w = \sqrt{24-70i}$. We will first find $w$.
Let $w = x+iy$.
$w^2 = (x+iy)^2 = (x^2-y^2) + i(2xy) = 24 - 70i$.
Comparing real and imaginary parts:
1. $x^2 - y^2 = 24$
2. $2xy = -70 \implies xy = -35$
We also know that $|w|^2 = |w^2|$, so $(x^2+y^2) = |24-70i|$.
$x^2+y^2 = \sqrt{24^2 + (-70)^2} = \sqrt{576 + 4900} = \sqrt{5476} = 74$.
3. $x^2 + y^2 = 74$
Now we have a system of two linear equations in $x^2$ and $y^2$:
(1) + (3): $2x^2 = 24 + 74 = 98 \implies x^2 = 49 \implies x = \pm 7$.
(3) - (1): $2y^2 = 74 - 24 = 50 \implies y^2 = 25 \implies y = \pm 5$.
From equation (2), $xy=-35$, which is negative, so $x$ and $y$ must have opposite signs.
The possible pairs for $(x,y)$ are $(7, -5)$ and $(-7, 5)$.
So, the two square roots of $24-70i$ are $7-5i$ and $-7+5i$.
Let's choose one of them, say $w = \sqrt{24-70i} = -7+5i$.
\ Now we need to find the values of $\sqrt{24-70i} + \sqrt{-24+70i}$.
Let the two square roots of $24-70i$ be $w_1 = 7-5i$ and $w_2 = -7+5i$.
The two square roots of $-24+70i$ will be $\pm i w_1$ or $\pm i w_2$.
Let's check: $i(7-5i) = 7i - 5i^2 = 5+7i$.
$(5+7i)^2 = 25 + 70i + 49i^2 = 25+70i-49 = -24+70i$. This is correct.
So the square roots of $-24+70i$ are $\pm(5+7i)$.
The expression has four possible values:
1. $(7-5i) + (5+7i) = 12 + 2i$
2. $(7-5i) - (5+7i) = 2 - 12i$
3. $(-7+5i) + (5+7i) = -2 + 12i$
4. $(-7+5i) - (5+7i) = -12 - 2i$
Step 4: Final Answer
We check the options against our four calculated values. The value $-12-2i$ is present in option (D).