Expert: Paul Klarreich - 2/11/2011

Question
The straight line x+y=a will be a tangent to the ellipse (x^2/9 + y^2/16)= 1 if a is equal to

Answer
Questioner: shakil
Country: India
Category: Calculus
Private: No
Subject: tangent and normal
Question: The straight line x+y=a will be a tangent to the ellipse (x^2/9 + y^2/16)= 1 if a is equal to
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x + y = a  has slope -1.
So you want a point where the tangent line has that slope.

Use implicit diff:

2x/9 + 2y y'/16 = 0

x/9 + y y'/16 = 0

y y'/16 = - x/9

y' = (-x/9)/(y/16)

y' = -16x/9y

Set that equal to - 1

-16x/9y = -1

16x = 9y

y = 16x/9

Now substitute back into  x^2/9 + y^2/16 = 1 and get your x and y.  Then you are practically done.  I'll leave that to you.