The positive value of a is 3
What are complex numbers?
Complex numbers are numbers that consists of a real number and an imaginary number. Complex numbers can be written in the form of
a + ib where a is the real part and ib is the imaginary part.
i=\sqrt{-1}
Representation of complex numbers:
- We know that a complex number of the form a + ib can be represented as point (a, b) in two dimensional plane.
Here, a + 7i and -8 + 4i are the given complex numbers
- a + 7i can be represented as (a ,7) in two dimensional plane.
- -8 + 4i can be represented as (-8, 4) in two dimensional plane.
It is given that the distance between a + 7i and -8 + 4i is √130 units
∴ We can say that the distance between (a ,7) and (-8, 4) is √130 units
How to find the distance?
The distance between 2 points any two points (p ,q) and (r ,s) can be found with the help of the formula
\sqrt{(r-p)^{2}+(s-q)^{2} }
∴ the distance between (a ,7) and (-8, 4) is \sqrt{(-8-a)^{2}+(4-7)^{2} }
Again, the distance between (a ,7) and (-8, 4) is √130 units
∴ \sqrt{(-8-a)^{2}+(4-7)^{2} } = √130
Now, squaring both sides, we get,
(-8-a)^{2} +9=130
⇒(-8-a)^{2}=121
Taking square root to both sides, we get
-8-a=±11
⇒a=-19 or 3
Here the positive value of a is asked
∴ a=3
The positive value of a is 3
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