Consider the following 11x2 24xy 4y 12 0 Use the discrim

Consider the following 11x^2 - 24xy + 4y + 12 = 0 Use the discriminated to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola parabola ellipse hyperbola Use a rotation of axes to eliminate the xy-term. (Write an equation in xy-coordinates. Use a rotation angle that satisfies 0 lessthanequalto phi lessthanequalto pi/2.) sketch the graph.

Solution

A general second order equation in two variables is of the form

Ax² + 2Bxy + Cy² + Dx + Ey + F = 0

for real constants A, B, C, D, E, and F. If you evaluate the discriminant B² - AC, you can determine if the equation is parabolic, hyperbolic, or elliptic. The cases are

B² - AC > 0, hyperbolic
B² - AC = 0, parabolic
B² - AC < 0, elliptic

so given equation is 11x^2 -24xy +4y^2 +12=0

A=11,B=-12 C=4

B^2 -AC = (12)^2 - 11*4

=144 -44

=100

B^2-AC >0

so it is hyperbola

a). option 3 is answer

b). first graph