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?
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.