17x2+18xyβ7y2β16xβ32yβ18=0
The rotation angle:
tan2ΞΈ=AβCBβ
where A, B and C are the corresponding coefficients from the equation:
Ax2+Bxy+Cy2+Dx+Ey+F=0
tan2ΞΈ=1+718β=43β=1βtan2ΞΈ2tanΞΈβ
tanΞΈ=31β
The required transformation equations needed to rotate the coordinate axes to eliminate the xy
term in the given equation for the conic section are:
x=xβ²cosΞΈβyβ²sinΞΈ
y=xβ²cosΞΈ+yβ²sinΞΈ
cosΞΈ=3/10β,sinΞΈ=1/10β
x=10β3xβ²βyβ²β,y=10βxβ²+3yβ²β
Substituting these expressions for x and y into the original equation for the conic section and simplifying, we get:
20xβ²2β10yβ²2β10β80βxβ²β10β80βyβ²β18=0
Simplifying and completing the squares to obtain an equation for a hyperbola in standard form, we get:
1/2(xβ²β2/10β)2ββ1(yβ²+4/10β)2β=1
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