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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.

## Complex Number – ISI MStat Year 2015 PSA Question 18

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

### Key Concepts

Complex number representation

Straight line

But try the problem first…

Answer: is a pair of intersecting straight lines

Source

ISI MStat 2015 PSA Problem 18

Precollege Mathematics

## Try with Hints

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.