Find the equation of the sphere centered at (2,0,-3) that is tangent to the plane x=y.
What is the point of tangency? Describe the interior of the sphere with an inequality.
What I have thus far:
The Equation of a Sphere Format $(x-x_0)^2+(y+y_0)^2+(z+z_0)^2=(r)^2$
which is $(x-2)^2+(y)^2+(z+3)^2=(\sqrt2)^2$
I've tried finding the point by creating a parametric equation:
$r(t)=<2,0,-3>+t<1,-1,0>$ because the Equation of a Plane Format is $Ax+By+Cz+D=0$
So, x=y Plane is really $x-y=0$
With that, I have:
x(t)=2+t
y(t)=-t
z(t)=-3
I looked at similar problems but do not grasps all the steps, please be concise.
I proceeded with the advise point of tangency, but it doesn't correspond with my graph.
You're alright here, the line you've created passes through the center of the sphere and is perpendicular to the plane $x=y$ (as it should be, since the tangent planes of the sphere are perpendicular to the vectors pointing from the center to a point on the sphere's surface). Now you must find the point on that line which lies on a sphere with equation $(x-2)^2 + y^2 + (z+3)^2 = r^2$ and also lies on the plane $x = y$. This is the point of tangency.
My trouble is seeing why you've assumed the sphere has radius $\sqrt{2}$, which is not obvious to me. Leaving $r$ as a variable, we'll find the $r$ value which allows us to simultaneously solve $$ (x-2)^2 + y^2 + (z+3)^2 = r^2, ~~x=y, ~~ \textrm{ and } (x,y,z) = (2+t, -t, -3)$$ for some $t \in \mathbb{R}$. Plugging in our $x$, $y$, and $z$ values from the line's equation into the plane's equation we have $$ x = y \to 2+t = -t$$ which gives $t = -1$. Now we solve for $(x,y,z) = (2+t, -t, -3) = (1,1,-3)$. This point must be on the plane and your line, so this is the point of tangency! We choose $r$ in order to make this point also lie on the sphere:
$$r^2 = (x-2)^2 + y^2 + (z+3)^2 = (1-2)^2 + (-1)^2 + (-3+3)^2 = 1 + 1, $$ $$ r^2 = 2 $$
so $r = \sqrt{2}$ as you stated before. This gives us the equation for the sphere, and by rewriting it as $$ (x-2)^2 + y^2 + (z+3)^2 \leq 2$$ we find an inequality describing the sphere's interior.