Question

The set of points z satisfying $|z+4|+|z-4|=10$ is contained in or equal to

• A. An ellipse with eccentricity $\frac{4}{5}$
• B. The set of points z satisfying Im(z) = 0
• C. The set of points satisfying Im(z)<=1
• D. Straight line
• E. None of the above

Answer 1

The correct option is A An ellipse with eccentricity $\frac{4}{5}$

Steps for simplification are:

Insert the value of $Z$ as $x+iy$ and apply the magnitude formula of the complex numbers: $\sqrt{(x^2+y^2)}$

Take the part obtained from $|z+4|$ to the $RHS$ and then square both the sides; you will get on simplification

$\sqrt{(x+4{)}^2+(y{)}^2}+\sqrt{(x-4{)}^2+(y{)}^2}=10$

$⟹\sqrt{(x-4{)}^2+y^2}=10-\sqrt{(x+4{)}^2+y^2}$

Square both sides:

$x^2+y^2+16-8x=100+x^2+y^2+16+8x-20\sqrt{(x+4{)}^2+y^2}$

Removing common terms and common factors:

$4x+25=5\sqrt{(x+4{)}^2+y^2}$

Again square both sides and then simplify to obtain the equation

$16x^2+625+200x=25x^2+25y^2+200x+400$

$⟹9x^2+25y^2=225$

It is equation of an ellipse.

So, $e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{9}{25}}=\frac{4}{5}$