   Chapter 10.5, Problem 58E

Chapter
Section
Textbook Problem

# Show that if an ellipse and a hyperbola have the same foci, then their tangent lines at each point of intersection are perpendicular.

To determine

To Show: The ellipse and a hyperbola have same foci, and then their tangent lines at each point of intersection are perpendicular.

Explanation

Calculation:

Write the equations of curve.

B2(x2A2y2B2)+b2(x2a2+y2b2)=B2+b2 (1)

Calculate the value of x from the equation (1).

B2(x2A2y2B2)+b2(x2a2+y2b2)=B2+b2B2x2A2+b2x2a2=B2+b2x2(B2A2+b2a2)=B2+b2x2=B2+b2(B2A2+b2a2)x2=B2+b2B2a2+b2A2A2a2x2=A2a2(B2+b2)B2a2+b2A2

Similarly, write the equation for variable y as below.

y2=B2b2(a2A2)b2A2+a2B2

Find the slope of the tangent lines of the curve for ellipse curve.

x2a2+y2b2=1

Differentiate the above equation with respect to x.

x2a2+y2b2=12xa2+2y.y'b2=02y.y'b2=2xa2yy'b2=xa2yE'=xb2ya2

Find the slope of the tangent lines of the curve for hyperbola curve.

x2A2y2B2=1

Differentiate the above equation with respect to x.

x2A2y2B2=12xA22y.y'B2=02y.y'B2=2xA2yy'B2=xA2yH'=xB2yA2

Calculate the product of the slope

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