GEC BHAVNAGAR

O

rthog

onal

Tra

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• An orthogonal trajectory of a family of curves is a curve that intersects each curve of the family orthogonally—that is, at right angles.

Ort

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METHODMETHODGiven family of curve ( , , )F x y c o

STEP 1 Find the differential equation for the given family of curves, by differentiating Eq.1.

( , )dy f x ydx

STEP 2 Find the differential equation of the Orthogonal Trajectory

1( , )

dydx f x y

STEP 3

Eq.3.

Eq.2.

Eq.1.

Solution of Eq.3. will be the equation of the family of orthogonal trajectories

( , , )G x y c o

Orth

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Chap

ter 3

Orthogonal Curves (1)

By differentiation we get: . Hence the family of parabola

in question satisfies the differential equation 2 .

2

dy xdx

dy xdx

2Consider the family of parabola . Find the family of curveswhich intersect the above family of parabola perpendicularly.

y x C Example

Solution

Two curve intersect perpendicularly if the product of the slopes of the tangents at the intersection point is -1.

The differential equation for the orthogonal family of curves.

1

2dydx xOr

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r 3

Orthogonal Curves (2)

1 1 1

2 2dy dy dxdx x x

It remains to solve

2Consider the family of parabola . Find the family of curveswhich intersect the above family of parabola perpendicularly.

y x C Example

Solution (cont’d)

The figure on the right shows these two orthogonal families of curves.

1

2dydx x

1 ln2

y x C

Orth

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Chap

ter 3

Describe the orthogonal trajectories for the family of curves given by

xy = C

Solution

y2 – x2 = K

y2 – x2 = K

Find the orthogonal trajectories of the family of curves x = ky2, where k is an arbitrary constant.

11 2 or = 2

dy dykydx dx ky

2

1 12 2

dyxdx ky yy

Solution

2dy ydx x

2dy xdx y

22

22

2

2

2

y dy x dx

y x C

yx C