WHat is the slope and are there any points where the slope is not defined?

Question

Find the slope of the tangent line to the ellipse x^2/16 +y^2/4 = 1 at the point (x,y). slope = ?

Are there any points where the slope is not defined? (Enter them as comma-separated ordered-pairs, e.g., (1,3), (-2,5). Enter none if there are no such points.) slope is undefined at

Answer 1

The first step is to write the equation of the ellipse as y(x)=±2(1-x^2/16)^(1/2) where the positive sign refers to the part of the ellipses which lies in the upper part of the plane and the negative sign refers to the part of the ellipses which likes in the lower part of the plane.

The derivative of the above equation becomes

dy/dx = ±(1-x^2/16)^(-1/2)(-2/16 x)

In this way it is easy to see that the derivative is always defined unless 1-x^2/16=0 (which is when the denominator of the derivative is equal to zero). The last condition implies x=±4 which are the points at which the derivative is not defined. These points corresponds to the intersection between the ellipses and the x-axe for which the tangent line to the ellipse becomes parallel to the y-axe (infinite slope).

So finally the points at which the derivative is not defined are p1=(-4,0) and p2=(4,0)

Best,

Davide