Question

True or False

The equation $a{x}^2+6y^2-12y=0$defines an ellipse if width="36">a>0

Short answer

The statement is true.

Step-by-step solution

Step 1. Given information

An equation is$a{x}^2+6y^2-12y=0$.

Step 2. State whether the statement is true or false.

The ellipse is given by the equation given below.

$a{x}^2+6y^2-12y=0$

The ellipse is characterized by negative value of $B^2-4AC$.

$\begin{array}{rcl}{B}^2-4AC& =& {\left(0\right)}^2-4\left(a\right)\left(6\right)\\ & =& -24a\end{array}$

The range of $a$can be determined for which the given equation is an ellipse.

$\begin{array}{rcl}{B}^2-4AC& <& 0\\ -24a& <& 0\\ a& >& 0\end{array}$

The given equation is an ellipse if for $a>0$, therefore the statement is True.