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.
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 :
$\displaystyle 3x^2+2\left(-\frac{3x}2\right)^2=12$
$\displaystyle 3x^2+\frac{9x^2}{2}=12$
$\displaystyle x^2+\frac{3x^2}{2}=4$
$\displaystyle \frac{2x^2+3x^2}{2}=4$
$\displaystyle 5x^2=8$
$\displaystyle \implies x=\pm \dots$