I am asked to find the type equation $\sqrt{x}+\sqrt{y} = \sqrt{a}$ , represents ? i.e a parabola , or hyperbola or ellipse or circle by squaring twice?
Now , what I have done is like this
$$\sqrt{x}+\sqrt{y} = \sqrt{a}$$ $$or,\ x+y-a = -2\sqrt{xy}\ (squaring)$$ $$or,\ x^2+y^2+a^2-2ax-2ay-2xy = 0\ (squaring\ again)---(1)$$
Now I think this is the equation of a circle , but if I plot the equation $y= x-2 \sqrt{x}+1$ in graph which is obtained by squaring once , I get half of a parabola.
again if I differentiate (1) I get $$2x+2y\frac{dy}{dx}-2a-2a\frac{dy}{dx}-2(x\frac{dy}{dx}+y) = 0$$
Now , if i put $\frac{dy}{dx}=0$ ,to maximize the function I get $2x-2y-2a=0$ , How the maximum point is the equation of a straight line in 2 dimension? there is no z component here !! can someone clear up a bit ?
$ax^2 + bxy + c^2 = 0$ is a parabola if $b^2 = 4ac$ the curve $\sqrt x + \sqrt y = \sqrt a$ is part of a parabola because you have $x^2 - 2xy + y^2 + \cdots$ and the discriminant $(-2)^2 - 4*1*1 = 0.$