   Chapter 3.5, Problem 73E

Chapter
Section
Textbook Problem

The equation x2 – xy + y2 = 3 re presents a "rotated ellipse,” that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.

To determine

To find: The points when the ellipse crosses the x-axis and to show that the tangent line at these points are parallel.

Explanation

Given:

The equation of the ellipse x2xy+y2=3.

Derivative rules:

Product rule:ddx(fg)=fddx(g)+gddx(f)

Chain rule: dydx=dydududx

Calculation:

Obtain the points if the ellipse crosses the x-axis.

The ellipse crosses the x-axis. That is, y=0.

Substitute y=0 in the equation x2xy+y2=3,

x20+0=3x2=3x=±3

Therefore, the ellipse crosses the x-axis at (3,0) and (3,0).

Obtain the slope of the tangent at the points (3,0) and (3,0).

Differentiate x2xy+y2=3 implicitly with respect to x.

ddx(x2xy+y2)=ddx(3)ddx(x2)ddx(xy)+ddx(y2)=02xddx(xy)+ddx(y2)=0

Apply the product rule and the chain rule

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