Find the Lines that are tangent and normal to curve at given point.

Jul 2008
27
0
san antonio Tx
Find the Lines that are Tangent and Normal to the Curve at the given points.


x^2+xy-y^2 = 11 , (3,1)


a) Give the equation that is tangent to the curve.
2x+(x dy/x + y dy/x) - 2y dy/dx = 0
(x-2y)dy/dx + 2x+y = 0
(x-2y)dy/dx = -2x-y
dy/dx = -2x-y/x-2y

dy/dx= -2(3)-1 / 3-2(1) = 7/1

y= y1 + m(x-x1)
y= 1 + 7/1(x-3) = ??? (I have tried many problems like this in Mymathlab and always get wrong this part)


b) Give the Equation that is Normal to the Curve.

y=1-7/1(x-3)

y= (also can't get this part right)


Please if Someone can show step by step how to do this problem I will appreciated.
 

Plato

MHF Helper
Aug 2006
22,473
8,643
Find the Lines that are Tangent and Normal to the Curve at the given points.


x^2+xy-y^2 = 11 , (3,1)
If \(\displaystyle m\ne 0\) is the slope of the curve at \(\displaystyle (p,q)\) on the curve:
The tangent is \(\displaystyle y-q=m(x-p)\) and the normal is \(\displaystyle y-q=\frac{-1}{m}(x-p).\)