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This is a beautiful problem from ISI MSTAT 2015 PSA problem 18 based on complex number. We provide sequential hints so that you can try.

The set of complex numbers $z$ satisfying the equation \( (3+7 i) z+(10-2 i) \bar{z}+100=0\) represents, in the complex plane

- a straight line
- a pair of intersecting straight lines
- a point
- a pair of distinct parallel straight lines

Complex number representation

Straight line

But try the problem first...

Answer: is a pair of intersecting straight lines

Source

Suggested Reading

ISI MStat 2015 PSA Problem 18

Precollege Mathematics

First hint

Simplify the Complex. Just Solve.

Second Hint

Let \(z = x+iy, \bar{z} = x-iy\) Then the given equation reduces to \((13x-9y+100)+i(5x-7y) = 0\).

Which implies \(13x-9y+100 = 0, 5x-7y = 0\).

They do intersect.(?)

Final Step

Yes! they intersect and to get the point of intersection just use substitution . Hence it gives a pair of intersecting straight lines.

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