Question

Given that 4y^2+9x^2+16y+18x=k is an ellipse equation. Find the value of k.
I want the steps. I want to know how to solve it. Thank you.

Answer 1

We gather complete squares in left part of equality

$$\begin{array}{l} 4 y ^ {2} + 9 x ^ {2} + 16 y + 18 x = k \\ 4 (y ^ {2} + 4 y + 4) - 16 + 9 (x ^ {2} + 2 x + 1) - 9 = k \\ 4 (y + 2) ^ {2} + 9 (x + 1) ^ {2} = k + 25 \\ \end{array}$$

So, this equation must describe an ellipse. For any $k > -25$ we can perform the following:

$$\begin{array}{l} \frac {4 (y + 2) ^ {2}}{k + 25} + \frac {9 (x + 1) ^ {2}}{k + 25} = 1; \\ \frac {\left(y + 2\right) ^ {2}}{\frac {k + 25}{4}} + \frac {\left(x + 1\right) ^ {2}}{\frac {k + 25}{9}} = 1; \\ \frac {\left(y + 2\right) ^ {2}}{\left(\sqrt {\frac {k + 25}{4}}\right) ^ {2}} + \frac {\left(x + 1\right) ^ {2}}{\left(\sqrt {\frac {k + 25}{9}}\right) ^ {2}} = 1; \\ \end{array}$$

The last equation describes an ellipse with semi-axes $\sqrt{\frac{k + 25}{4}}, \sqrt{\frac{k + 25}{9}}$ , because of $k > -25$ , they have positive values. Otherwise this values would be zero or even complex, that's why this equation would not be an equation of an ellipse.