Question

How do you identify if the equation $9{{x}^{2}}+4{{y}^{2}}-36=0$ is a parabola, ellipse, or hyperbola and how do you graph it?

Answer 1

The general equation of a conic is $a{{x}^{2}}+b{{y}^{2}}+2hxy+2gx+2fy+c=0$. We can determine whether the given conic is a circle, ellipse, or hyperbola from the equation of the conic. For the conic of the equation $a{{x}^{2}}+b{{y}^{2}}+2hxy+2gx+2fy+c=0$,
If $a=b\And h=0$, then the conic is a circle. If $\Delta =4({{h}^{2}}-ab)$ is negative, then the conic is an ellipse. If the $\Delta =4({{h}^{2}}-ab)$ is positive then the conic is a hyperbola.

Complete step by step answer:
The given equation of conic is $9{{x}^{2}}+4{{y}^{2}}-36=0$. Comparing with the general equation of the conic $a{{x}^{2}}+b{{y}^{2}}+2hxy+2gx+2fy+c=0$, we get $a=9,b=4,c=-36,\And h=g=f=0$.
Let’s verify the conditions for the circle, ellipse, and hyperbola.
Here as $a\ne b$ the conic of the equation is not a circle.
$\Delta =4({{h}^{2}}-ab)$, substituting the value of coefficients, we get

& \Rightarrow \Delta =4\left( {{0}^{2}}-9\times 4 \right) \\\ & \Rightarrow \Delta =4\left( -36 \right) \\\ \end{aligned}$$ Multiplying 4 and $$-36$$ equals $$-144$$, substituting above we get $$\Rightarrow \Delta =-144$$ Hence, as $$\Delta $$ is a negative value the equation is of an ellipse. To graph the ellipse, we first have to rearrange the given equation, as follows $$9{{x}^{2}}+4{{y}^{2}}-36=0$$ Adding 36 to both sides of the equation, we get $$\Rightarrow 9{{x}^{2}}+4{{y}^{2}}-36+36=0+36$$ $$\Rightarrow 9{{x}^{2}}+4{{y}^{2}}=36$$ Dividing both sides of the above equation by 36, we get $$\Rightarrow \dfrac{9{{x}^{2}}+4{{y}^{2}}}{36}=\dfrac{36}{36}$$ The above equation can also be written as $$\Rightarrow \dfrac{9{{x}^{2}}}{36}+\dfrac{4{{y}^{2}}}{36}=1$$ Canceling out common factors from the left-hand side of the equation, we get $$\Rightarrow \dfrac{{{x}^{2}}}{4}+\dfrac{{{y}^{2}}}{9}=1$$ We can graph the ellipse $$\dfrac{{{x}^{2}}}{4}+\dfrac{{{y}^{2}}}{9}=1$$ as follows  **Note:** To identify the conic from the equation, one should remember the different properties & conditions of the conics. One can make calculation mistakes while finding the coefficient of the $$x,y\And xy$$. It should be remembered that in the general equation their coefficients are $$2g,2f\And 2h$$ respectively. Hence, we have to half the coefficients of $$x,y\And xy$$to find the values of $$g,f\And h$$ and then use them to verify conditions.