Find the shortest distance from the origin to the hyperbola $x^2+8xy+7y^2=225$
i know that
$$d(x_0, x) = \sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}$$
I also found this formula in my notes
$$ d(x_0,p) = \frac{|ax_0+by_0+cz_{0}-c|}{\sqrt{a^2+b^2+c^2}}$$
I just haven't seen it been applied in class so i'm a bit confused.
This was in a week we were learning about Lagrange Multipliers and we don't seem to be given a constraint.
Best Answer
let $P(rcost,rsint)$ be a Point on Hyperbola. so its distance from $(0,0)$ is $r$, so we need to find Minimum value of $r$. Since $P$ lies on Hyperbola
$$r^2cos^2t+8r^2sintcost+7r^2sin^2t=225$$ $\implies$
$$r^2=\frac{450}{8sin2t-6cos2t+8}$$
Now max value of $$8sin2t-6cos2t+8$$ is $$\sqrt{8^2+6^2}+8=18$$
hence Min value of $r^2$ is $\frac{450}{18}=25$
So shortest distance is $5$