For the equation, x^2 + 4xy - 2y^2 - 7 = 0, answer the...

For the equation, $x^2 + 4xy - 2y^2 - 7 = 0$, answer the following questions. a. Rewrite the equation in a rotated $x'y'$-system without an $x'y'$-term. Use the appropriate rotation formulas. b. Express the equation involving $x'$ and $y'$ in the standard form of a conic section. c. Use the rotated system to graph the system. a. The equation in a rotated $x'y'$-system is

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00:01 From exercise 14 we have x is equal to 3x dash minus 4 y dash divided by 5 and y is equal to 4 x dash plus 3 y dash divided by 5 we have to replace the value of x and y in equation 32 x square minus 48 x y plus 18 y square minus 15 x minus 20 y is equal to 0 and then we have to find the equation in x -y -ddice -y -dice only now replace them we get 32 multiplied by 3x dash minus 4 y -dess divided by 5 wall square minus 48 multiplied by 3x dash minus 4 y -d s divided by 5 multiplied by 4x dash plus 3 y -dess divided by 5 plus 18 4x dash minus 3 y divided by 5 whole square minus 15 multiplied by 3x dash minus 4 y dash divided by 5 minus 20 multiplied by 4x dash plus 3 y dash divide by 5 equal to 0 after simplification we get 32 multiplied by 9 x -dose square minus 24 x -dess y -dess plus 16 y -dess square divided by 25 minus 48 multiplied by 12 x -dice square minus 7 x -y -dess y -dus minus 12 y -dus square divided by 25 plus 18 multiplied by 16 x -dase square plus 24 x -d s y -d s plus 9 y -dash squared divided by 25 minus 25 x -des equal to 0 after simplification and multiply both side by 25 we get 288 x -d square minus 768 x -d dash y -dice plus 512 y -12 square minus 576 x dash square plus 336 x -desquare plus 536 x -desquare plus 576 y -dash square plus 288 x -dash -square plus 432 x -des -y -dice plus 162 y -dask square minus 625 x -des equal to 0 after simplification we get 1 ,250 y -d s square minus 625 x -x -dis equal to 0 this is the equation in x -y -dess y -dess form now for part b we simplify this we get 12 1 ,250 divided by 1 ,250 y -dice square is equal to 625 divided by 1 ,250 y -dash -square is equal to 625 divided by 1 ,000 50 multiplied by x -dash after simplification we get y -dash square is equal to 1 divided by 2 x -dash this represents a parabola whose axis is x -dash now we will draw the graph of this parabola let's say this is x -axis and this is y -axis we know the angle of rotation is 53 degree let's say this is equal to 53 degree and this line is x -dash -axis and this line is y -dash line...

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