Question

Identify the type of surface represented by the given equation. x2 z2 y 14) 5 7 5

Answer 1

Solution: Given a surface:

$\Rightarrow \frac{x^2}{5}+\frac{z^2}{7}=\frac{y}{5}$

$\Rightarrow \frac{x^2}{1}+\frac{z^2}{\frac{7}{5}}=y$

The standard equation of a paraboloid with origin as its vertex and z-axis as its axis is:

$\Rightarrow \frac{x^2}{a^2}+\frac{y^2}{b^2}=z$

There are other forms of the standard equation of the paraboloid with different axis:

$\Rightarrow \frac{x^2}{a^2}+\frac{z^2}{b^2}=y$

$\Rightarrow \frac{y^2}{a^2}+\frac{z^2}{b^2}=x$

Basically in the equation of a paraboloid, the variable having the power 1 will be the axis of the paraboloid, therefore, in the equation (1), since the variable y is having the power 1, therefore, it is a paraboloid with y-axis as its axis and origin as its vertex. Therefore, its graph will be:

6 z-axis 5 فع -8 2 y-axis x-axis -3

Therefore, the surface given in the question is a paraboloid.

I hope it helps you!