Question

In the following exercises, identify the type of graph.

$$\begin{gathered} \left(a\right)x^2+y^2+4x-10y+25=0 \\ \left(b\right)y^2-x^2-4y+2x-6=0 \\ \left(c\right)x=-y^2-2y+3 \\ \left(d\right)16x^2+9y^2=144 \end{gathered}$$

Short answer

The equation represents

(a) Circle

(b) Hyperbola

(c) parabola

(d) ellipse

Step-by-step solution

Step 1. Given Information

The given equations are

$$\begin{gathered} \left(a\right)x^2+y^2+4x-10y+25=0 \\ \left(b\right)y^2-x^2-4y+2x-6=0 \\ \left(c\right)x=-y^2-2y+3 \\ \left(d\right)16x^2+9y^2=144 \end{gathered}$$

Part (a) Step 1. Explanation

From the given equation it is very clear that $x^{2}and{y}^2$ terms have same sign and same coefficient. So, the equation represents the Circle.

Part(b) Step 1. Explanation

From the given equation it is very clear that $x^{2}and{y}^2$terms have different sign and different coefficient. So, the equation represents the hyperbola.

Part(c) Step 1. Explanation

From the given equation it is very clear that one variable $y$ is squared. So, the equation represents the Parabola.

Part(d) Step 1. Explanation

From the given equation it is very clear that $x^{2}and{y}^2$ terms have same sign and different coefficient. So, the equation represents an ellipse.