Points on the curve

Question

At what point(s) on the curve x2 + y2 = 9 is the tangent line vertical? (2 points)

(-3, 0) (3, 0)

(-9, 0) (9, 0)

(0, 9) (0, -9)

(0, 3) (0, -3)

Answer 1

Hi Gail,

 

A vertical line has a slope that is undefined. So what we want to know is where the first derivative is undefined. Differentiating the equation

                                         x²  +  y²  =  9  results in  2x + 2y y’  =  0  and solve for y’.

 

Since y’ =  -x/y and a vertical line has a slope that is undefined, we consider what values make the fraction undefined. When y = 0 the fraction -x/y is undefined, y has to be zero. Substituting y = 0 into the original equation results in 

 

                     x²  +  0²  = 9  and solve for x.  In this case there are two values of x which are 3 and -3.

 

Therefore the points at which the tangent lines are vertical are (3.0) and (-3,0)