Chapter 8.4, Problem 64E

Solution

a.

To find the change in sign of the discriminant $B^2-4AC$ when the axes are translated $h$ unit horizontally and $k$ units vertically.

The sign of discriminant $B^2-4AC$ remains invariant.

Given information:

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

Calculation:

Assume the graph of the equation is shifted $h$ units to the left and moved $k$ units up. So, the new equation will be $A{\left(x+h\right)}^2+B\left(x+h\right)\left(y+k\right)+C{\left(y+k\right)}^2+D\left(x+h\right)+E\left(y+k\right)+F=0$ .

Simplify the equation as follows:

  $\begin{array}{c}A{\left(x+h\right)}^2+B\left(x+h\right)\left(y+k\right)+C{\left(y+k\right)}^2+D\left(x+h\right)+E\left(y+k\right)+F=0\\ A\left(x^2+2hx+h^2\right)+B\left(xy+hy+kx+hk\right)+C\left(y^2+2ky+k^2\right)+D\left(x+h\right)+E\left(y+k\right)+F=0\\ A{x}^2+Bxy+C{y}^2+A\left(2hx+h^2\right)+B\left(hy+kx+hk\right)+C\left(2ky+k^2\right)+D\left(x+h\right)+E\left(y+k\right)+F=0\end{array}$

Observe that coefficient of $x^2$ , $xy$ and $y^2$ remain same. So, the value of the discriminant $B^2-4AC$ will remain same. So, there is no change in the value of the discriminant by horizontal and vertical translation.

Conclusion:

The sign of discriminant $B^2-4AC$ remains invariant.

b.

To find the change in sign of the discriminant $B^2-4AC$ when both sides of the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ are multiplied by the same nonzero constant $k$ .

The sign of discriminant $B^2-4AC$ remains invariant.

Given information:

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

Calculation:

Multiply both sides of the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ by nonzero constant $k$ .

  $\begin{array}{c}k\left(A{x}^2+Bxy+C{y}^2+Dx+Ey+F\right)=k\cdot 0\\ kA{x}^2+kBxy+kC{y}^2+kDx+kEy+kF=0\end{array}$

Let $A\text{'}=kA$ , $B\text{'}=kB$ and $C\text{'}=kC$ .

Then, the new value becomes:

  $\begin{array}{c}B{\text{'}}^2-4A\text{'}C\text{'}={\left(kA\right)}^2-4\left(kA\right)\left(kC\right)\\ =k^2\left(B^2-4AC\right)\end{array}$

Since $k^2$ is always positive, the sign of $B^2-4AC$ is not affected by the multiplication.

Conclusion:

The sign of the discriminant $B^2-4AC$ remains invariant.