Question

Determine the conic section given by each of the following equations. Be sure to showall work to find the standard form of the equation of each conic section.1. 3x 2 + 5y 2 − 12x + 30y = −42

Answer 1

The standard form of eqution of a conic section is;

`\begin{gathered} Ax^2+Bxy+Cy^2+Dx+Ey+F=0 \\ \\ \text{ Where;} \\ A,B,C,D,E,F\text{ are real numbers;} \\ \\ A0,B\\e0,C\\e0 \end{gathered}`

Given the equation of the conic section;

`3x^2+5y^2-12x+30y=-42`

Hence;

`A=3,C=5,B=0`

Then;

`\begin{gathered} B^2-4AC=(0)^2-4(3)(5) \\ \\ B^2-4AC=-60 \\ \\ B^2-4AC<0 \\ \\ because\text{ }-60<0 \end{gathered}`

Thus;

`B^2-4AC<0,\text{ then the conic section is an ellipse}`

ANSWER: The equation given is an ellipse.

The standard form of an ellipse is;

`((x-h)^2)/(a^2)+((y-k)^2)/(b^2)=1`
`\begin{gathered} ((x-2)^2)/((√(5))^2)+((y-(-3))^2)/((√(3))^2)=1 \\ \\ ((x-2)^2)/(5)+((y+3)^2)/(3)=1 \end{gathered}`