73x^2-72xy+52y^2+30x+40y-75=0 use rotation of axis to eliminate the xy term in each of the following equations and describe the locus of conic section and sketch the graph?

Answer 1

To eliminate the xy term, we need to rotate the axes by an angle of θ such that the new axes are parallel to the axis of the conic section. The angle θ can be found using the formula:

tan(2θ) = (2AxBy)/(A^2-B^2)

Where A=73, B=52 and therefore tan(2θ) = (-2*73*52)/(73^2-52^2) = -6016/5761
which means
θ = 1/2 * tan^(-1)(-6016/5761) = 1.197 radians or approximately 68.6 degrees

Once we rotate the axes by 68.6 degrees, the equation becomes:

73x'^2 - 20y'^2 + 30x' + 40y' - 75 = 0

This equation represents a hyperbola, since the signs of the x'^2 and y'^2 terms are different. The locus of this conic section will be a hyperbola with its center at (-.205, 1.207) and with a vertical transverse axis.

Please note that the final step to plot the graph requires knowledge of software or graphing calculator to accurately draw the rotated hyperbola.