Question

The locus of a point in a Argand plane that moves satisfying the condition $|z-1+i|-|z-2-i|=3$ is?

• A. A circle with radius $3$ & centre at $z=\frac{3}{2}$
• B. An ellipse with its foci at $1-i$ & $2+i$ & major axis $=3$
• C. A hyperbola with foci at $1-i$ & $2+i$ & transversal axis $=3$
• D. None of the above

Answer 1

The correct option is C A hyperbola with foci at $1-i$ & $2+i$ & transversal axis $=3$

Point (v) of passage: If major axis is to be considered as transversal axis with length $2a<|z_1-z_2|$ then the equation of hyperbola is given by $||z-z_1|-|z-z_2||=2a$ where $z_1$ and $z_2$ are focii and transversal axis length as $2a$

Given equation: $|z-1+i|-|z-2-i|=3$

$⟹$ $|z-(1-i)|-|z-\left(\right(2+i)|=3$

This represents one half of a hyperbola with $(1-i)$ and $(2+i)$ as focii and transversal axis $=3$

Proof:

Let $z=x+iy$

$⟹$ $|z-(1-i)|-|z-\left(\right(2+i)|=3$

$⟹$ $|x+iy-(1-i)|-|x+iy-\left(\right(2+i)|=3$

$⟹$ $\sqrt{(x-1{)}^2+(y+1{)}^2}-\sqrt{(x-2{)}^2+(y-1{)}^2}=3$

$⟹$ $\sqrt{(x-1{)}^2+(y+1{)}^2}=3+\sqrt{(x-2{)}^2+(y-1{)}^2}$

$⟹$ $(x-1{)}^2+(y+1{)}^2=9+(x-2{)}^2+(y-1{)}^2+18\sqrt{(x-2{)}^2+(y-1{)}^2}$

$⟹$ $-6+x+2y=9\sqrt{(x-2{)}^2+(y-1{)}^2}$

Again take squares on both sides and after rearranging: Proceeding further we get equation of hyperbola