   Chapter 2.6, Problem 57E

Chapter
Section
Textbook Problem

# The equation x 2 − x y +   y 2 =   3 represents 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:

i. Points where ellipses cuts x axis

ii. To show tangent lines are parallel

Explanation

1) Formula:

i. Derivative of constant

ddxc=0,

ii. Product rule

ddxfx*gx=fxddxgx+gxddx fx

Given: x2-xy+y2=3

Calculation:

i.

When graph crosses x-axis, y coordinate becomes zero.

Substitute y=0 in given equation,

x2-x(0)+02=3

x2=3

Taking square root on both sides,

x2=±3

x=±3

Therefore the points when the curve crosses x- axis are -3,0 and 3,0

ii.

Differentiating given expression implicitly with respect to x on both sides,

ddxx2-xy+y2=ddx3

2x-xdydx+y1+2ydydx=0

2x-xdydx-y+2ydydx=0

-xdydx+2ydydx+2x-y=0

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