Let z=x2y2−1 and let C be the curve of intersection of the surface with the plane  y=2 . Find equations for the line tangent to C at the point P(1,2,13).

Respuesta :

Answer:

[tex]x=1+t,y=2,z=\frac{1}{3}+\frac{2}{3}t[/tex]

Step-by-step explanation:

[tex]z=\frac{x^2}{y^2-1}[/tex]

Point,P=(1,2,1/3)

Let x=t

y=2

[tex]z=\frac{t^2}{2^2-1}=\frac{t^2}{3}[/tex]

[tex]r(t)=<t,2,\frac{t^2}{3}>[/tex]

[tex]r'(t)=<1,0,\frac{2}{3}t>[/tex]

Substitute x=t=1

[tex]r(1)=<1,2,\frac{1}{3}>[/tex]

[tex]r'(4)=<1,0,\frac{2}{3}>[/tex]

The vector equation for the tangent line to C at the point P(1,2,1/3) is given by

[tex]r(t)=r(1)+tr'(1)[/tex]

[tex]r(t)=<1,2,\frac{1}{3}>+t<1,0,\frac{2}{3}>[/tex]

[tex]r(t)=<1+t,2,\frac{1}{3}+\frac{2}{3}t>[/tex]

Parametric equations of the tangent line to C at point P(1,2,1/3)

[tex]x=1+t,y=2,z=\frac{1}{3}+\frac{2}{3}t[/tex]