Find the equation of the locus of the point representing z if |z-5-12i| + |z-5+21i|=13
correction:
|z-5-12i| + |z-5+12i|=13
|z-5-12i| is the distance of z from 5+12i
|z-5+12i| is the distance of z from 5-12i
As we know, the locus of a point whose distances from two other points add to a constant, is an ellipse.
The equation of the ellipse should be no problem, since you know the foci and the sum of the distances.
To find the equation of the locus of the point representing z in the given equation, let's simplify the expression step by step.
First, using the definition of absolute value for complex numbers, |z| = sqrt(x^2 + y^2), where z = x + yi.
In the given equation, we have:
|z-5-12i| + |z-5+21i| = 13
Let's substitute z = x + yi into the equation:
| (x + yi) - 5 - 12i | + | (x + yi) - 5 + 21i | = 13
Simplifying the equation, we get:
|(x - 5) + (y - 12)i| + |(x - 5) + (y + 21)i| = 13
Now, let's compute each absolute value separately:
Let A = (x - 5) + (y - 12)i
Let B = (x - 5) + (y + 21)i
|A| = sqrt( (x - 5)^2 + (y - 12)^2 )
|B| = sqrt( (x - 5)^2 + (y + 21)^2 )
Substituting these values back into the equation, we have:
sqrt( (x - 5)^2 + (y - 12)^2 ) + sqrt( (x - 5)^2 + (y + 21)^2 ) = 13
This is the equation of the locus of the point representing z.
To find the equation of the locus of the point representing z, we can first simplify the given equation involving complex numbers.
Let's begin by using the definition of the absolute value of a complex number:
|a + bi| = √(a^2 + b^2)
We have:
|z-5-12i| + |z-5+21i| = 13
Applying the definition of absolute value, we get:
√((Re(z-5))^2 + (Im(z-5-12i))^2) + √((Re(z-5))^2 + (Im(z-5+21i))^2) = 13
Next, let's simplify further by expanding the square terms:
√((Re(z)-5)^2 + (Im(z-12i))^2) + √((Re(z)-5)^2 + (Im(z+21i))^2) = 13
Now, let's focus on each square root term separately.
For the first square root term:
√((Re(z)-5)^2 + (Im(z-12i))^2)
Using the definition of absolute value, we can rewrite (Im(z-12i))^2 as (Re(z))^2 + (-12)^2.
So, the first square root term simplifies to:
√((Re(z)-5)^2 + (Re(z))^2 + (-12)^2)
Simplifying further, we have:
√(2(Re(z))^2 - 10Re(z) + 25 + 12^2)
Similarly, for the second square root term:
√((Re(z)-5)^2 + (Im(z+21i))^2)
Using the definition of absolute value, we can rewrite (Im(z+21i))^2 as (Re(z))^2 + 21^2.
So, the second square root term simplifies to:
√((Re(z)-5)^2 + (Re(z))^2 + 21^2)
Simplifying further, we have:
√(2(Re(z))^2 - 10Re(z) + 25 + 21^2)
Now, we can rewrite the initial equation as:
√(2(Re(z))^2 - 10Re(z) + 25 + 12^2) + √(2(Re(z))^2 - 10Re(z) + 25 + 21^2) = 13
To simplify this equation even further, let's substitute a variable for Re(z):
Let x = Re(z)
Substituting this in, we have:
√(2x^2 - 10x + 169) + √(2x^2 - 10x + 676) = 13
Now, our task is to square both sides of the equation to eliminate the square roots.