FIND ALL POINTS ON THE CURVE WHERE THE SLOPE OF THE TANGENT LINE IS GIVEN

Example 1 :

Find the points on curve

x²-y²  =  2

at which the slope of the tangent is 2.

Solution :

slope of the tangent  =  2

m  =  2  ------(1)

x²-y²  =  2

2x-2y(dy/dx)  =  0

-2y(dy/dx)  =  -2x

dy/dx  =  x/y

Slope of the tangent drawn at the point on the curve 

=  x/y  ------(2)

(1)  =  (2)

x/y = 2

x  =  2y

By applying the value of x in the given curve, we get

If y = √(2/3), then x  =  2√(2/3)

If y = -√(2/3), then x  =  -2√(2/3)

So, the required points are

(2√(2/3), √(2/3)) and (-2√(2/3), - √(2/3))

Example 2 :

Find at what points on a circle

x²+y²  =  13

the slope of the tangent is -2/3.

Solution :

m  =  -2/3  ----(1)

x² + y²  =  13

2x+2y(dy/dx)  =  0

2y(dy/dx)  =  -2x

dy/dx  =  -x/y  --- (2)

(1)  =  (2)

-2/3 = -x/y

2y  =  3x

y  =  3x/2

By applying y = 3x/2 in the given equation, we get

x² + (3x/2)²  =  13

 x² + (9x²/4)  =  13

13x²/4  =  13

x²  =  13(4/13)

x  =  ± 2

If x = 2, then y = 3 

If x = -2, then y  =  -3

So, the required points are (2, 3) (-2, -3).

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