Consider the equation:2sqrt(3x)^2-6xy+sqrt(3x)+3y=0 a) Use the discriminant to determine

Rui Baldwin 2020-10-25 Answered

Consider the equation:
23x26xy+3x+3y=0
a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola?
b) Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies 0φπ/2)

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Expert Answer

Caren
Answered 2020-10-26 Author has 96 answers

Step 1
We have given an equation,
23x26xy+3x+3y=0
the general Cartesian form of a conic section:
Ax2+Bxy+Cy2+Dx+Ey+F=0
The discriminant is B24AC
(a) If B24AC<0, then the equation represents an ellipse.
(b) If B24AC=0, then the equation represents a parabola.
(c) If B24AC>0, then the equation represents a hyperbola.
Step 2
On comparing the given equation with the general form of conic section, we get,
A=23,B=6,C=0
The discriminant is 624×23×0=36
36>0
So the equation is hyperbola.
Step 3
b) In such a case, the relation between coordinate (x, y) and new coordinates (x', y') is given by:
x=xcosθysinθandy=xcosθ+ysinθ
cot2θ=CAB=0236
cot2θ=CAB=13
cot2θ=CAB=cot,π3
θ=π6
sin,π6=12
cos,π6=32
We shall find the value of x,y by these values.
x=xcosθysinθandy=xcosθ+ysinθ
x=x32y12andy=x32+y12
On plugging these values in the equation we get,
23x26xy+3x+3y=0
23(x32y12)26(x32y12)(x32+y12)+3(x32y12)+3(x32+y12)=0

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