Question

True or False

The equation $3x^2+Bxy+12y^2=10$defines a parabola if$B=-12$.

Short answer

The statement is true.

Step-by-step solution

Step 1. Given information

An equation is$3x^2+Bxy+12y^2=10$.

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

Put $B=-12$in the equation, $3x^2+Bxy+12y^2=10$.

$3x^2-12xy+12y^2=10$

Compare this equation with the general equation of a polynomial, containing two variables,

$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,$

Here,

$A=3,B=-12,C=12$

Now as any second degree polynomial, containing two variables, represents a parabola, if

$\begin{array}{rcl}{B}^2-4AC& =& 0\\ (-12{)}^2-4\times 3\times 12& =& 144-144\\ & =& 0\end{array}$

which is true.

So, the given equation represents a parabola, which is a true statement.