Question

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

Answer 1

Step 1 :

The tangent plane equation at the point is

The normal line to the surface at the point   is

Step 2 :

(a)

The equation is and the point is .

Rewrite the equation as .

Consider .

Differentiate partially with respect to x.

Substitute the point in above equation.

.

Differentiate partially with respect to y.

Substitute the point in above equation.

.

Differentiate partially with respect to z.

Substitute the point in above equation.

. 

Step 3 :

The equation of the tangent plane is

Substitute , and in above equation. 

The equation of a tangent plane is .

Step 4 :

(b)

Find the equation of normal line :

The equation of normal line is

Substitute , and in above equation.   

The equation of normal line is

Solution :    

(a) The equation of a tangent plane is .

(b) The equation of normal line is