Find all the points of the elispe 3x^2+2y^2=12 where the tangent line has slope 1. Make a sketch of the elispe and the tangent line to confirm your answer as well.

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- Jun 25th 2008, 01:34 PMlemonteaImplicit Differentitationl;finding equation
Find all the points of the elispe 3x^2+2y^2=12 where the tangent line has slope 1. Make a sketch of the elispe and the tangent line to confirm your answer as well.

- Jun 25th 2008, 01:52 PMMoo
Hello,

Differentiate :

$\displaystyle 6x+2(y^2)'=0$

Using the product rule :

$\displaystyle (y \cdot y)'=y'y+yy'=2yy'$.

Substituting :

$\displaystyle 6x+4yy'=0$

We want $\displaystyle y'=1$

--> $\displaystyle 6x+4y=0 \implies y=-\frac{3x}{2}$

Substituting in the equation of the ellipse :

$\displaystyle 3x^2+2\left(-\frac{3x}2\right)^2=12$

$\displaystyle 3x^2+\frac{9x^2}{2}=12$

$\displaystyle x^2+\frac{3x^2}{2}=4$

$\displaystyle \frac{2x^2+3x^2}{2}=4$

$\displaystyle 5x^2=8$

$\displaystyle \implies x=\pm \dots$