Triangle Inequality and Pseudometric

[fabbi007]

Homework Statement

$d(x,y)=(a|x_1-y_1|^2+b|x_1-y_1||x_2-y_2|+c|x_2-y_2|^2)^{1/2}$

where $a>0, b>0, c>0$ and $4ac-b^2<0$

Show whether $d(x,y)$ exhibits Triangle inequality?

Homework Equations

(M4) $d(x,y) \leq d(x,z)+d(z,y)$ (for all x,y and z in X)

The Attempt at a Solution

I started my solution by solving by squaring the both sides of the equation.

$d^2(x,y); [d(x,z)+d(z,y)]^2.$ separately

I am tending to think it does not satisfy the triangle inequality any other simple way to prove it? Also is this a pseudometric? if it does not satisfy the triangle inequality?

 

[fabbi007]

Any help is greatly appreciated!