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Worked solution

Eliminate a reciprocal parameter and sketch the hyperbola

A curve CC has parametric equations x=t3+2/t3x=t^3+2/t^3, y=t3−2/t3y=t^3-2/t^3, t∈Rt\in\mathbb R. Find a Cartesian equation of CC. Sketch it, giving exact axis intersections and equations of asymptotes.

[2013/CJC/MYE/5]

CJC · Discussion · Question 2

Answer

x2−y2=8x^2-y^2=8; x-axis points (±22,0)(\pm2\sqrt2,0), no y-axis crossing; asymptotes y=±xy=\pm x.

Eliminate t without losing its restriction

The reciprocals require t≠0t\ne0. Adding gives x+y=2t3x+y=2t^3; subtracting gives x−y=4/t3x-y=4/t^3. Multiplying gives (x+y)(x−y)=8(x+y)(x-y)=8, so x2−y2=8x^2-y^2=8. Conversely, every point on this hyperbola gives a nonzero real t3=(x+y)/2t^3=(x+y)/2, so both branches are traced.

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Identify the intersections and asymptotes

At y=0y=0, x=±8=±22x=\pm\sqrt8=\pm2\sqrt2, giving (±22,0)(\pm2\sqrt2,0). There are no y-axis intersections because x=0x=0 would give −y2=8-y^2=8. The asymptotes are y=xy=x and y=−xy=-x. Sketch two branches opening left and right, approaching these lines.

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Compare your sketch

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Hyperbola x squared minus y squared equals 8, with left and right branches, x-intercepts minus 2 square root 2 and plus 2 square root 2, no y-intercept, and dashed asymptotes y equals x and y equals minus x.
Hyperbola x squared minus y squared equals 8, with left and right branches, x-intercepts minus 2 square root 2 and plus 2 square root 2, no y-intercept, and dashed asymptotes y equals x and y equals minus x.
Hyperbola x squared minus y squared equals 8, with left and right branches, x-intercepts minus 2 square root 2 and plus 2 square root 2, no y-intercept, and dashed asymptotes y equals x and y equals minus x.

Common mistakes

The printed real-parameter statement still excludes zero because t³ occurs in a denominator.

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