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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Find all values of c such that the parabolas y = 4 x 2 and x = c + 2 y 2 interest each other at right angles.

To determine

To find:all the values of c

Explanation

1) Concept:

If two curves intersect perpendicular to each other at any point, then the tangents to the curve at that point are also perpendicular to each other. Therefore, product of slope of the tangent is – 1. Using this condition in given equations of curve, we can find value of c.

2) Formula:

Slope of tangent to curve y is

mtan=dydx

3) Given:

 Two curves y=4x2 and x=c+2y2 intersect each other at a point in the right angle.

4) Calculations:

Two curves y=4x2 and x=c+2y2 intersect each other at a point in right angle.

Therefore, tangent to given points are perpendicular to each other, that is, product of their slopes is -1.

So by definition of slope of tangent to the curve,

For y=4x2, slope  of tangent is

m1= dydx=8x

For x=c+2y2 after differentiating with respect to x we get,

1=4ydydx

Hence slope of tangent to the curve is

m2= dydx=14y

As tangents are perpendicular to each other, therefore, m1m2= -1

Hence

8x.14y= -1,       2x= -y

Squaring on both sides we get, 4x2=y2, but from equation of first curve  y=4x2, thus y=y2, 

y2-y=0, yy-1=0

y=0 or y=1,

When y = 0 ,2x= -y gives  x=0 

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