I have a hyperboloid $H$, which is $x^2+\frac{y^2}{4}-z^2=1$.
I want to show that all points $P$ on $H$, in which the tangentplane to $H$ that goes through the point $(1,4,2)$, all are in a plane. And thereafter also determine the equation of this plane.
How would I go by doing this. I'm unsure where to start?
$\endgroup$1 Answer
$\begingroup$Let $P(X,Y,Z)$.
Thus, the equation of our tangent plane it's: $$2X(x-X)+\frac{Y}{2}(y-Y)-2Z(z-Z)=0$$ or $$Xx+\frac{Y}{4}y-Zz-X^2-\frac{Y^2}{4}+Z^2=0$$ or since $(X,Y,Z)$ placed on our hyperboloid, we obtain $$Xx+\frac{Y}{4}y-Zz-1=0.$$ But $(1,4,2)$ placed on this plane, which gives $$X+Y-2Z-1=0,$$ which is an equation of the plane: $x+y-2z-1=0$.
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