<s>[INST] Distance between the curves from a particular point The distance between the points $P(u,v)$ and the curve $x^2+4x+y^2=0$ is the same as the distance between the points  $P(u,v)$ and $M(2,0)$. If $u$ and $v$ satisfy the relation $u^2-\frac{v^2}{q}=1$, then $q$ is greater than or equal to
(A) 1
(B) 2
(C) 3
(4) 4
My approach is as follow Curve $x^2+4x+y^2$ represent a circle with centre $(-2,0)$ and radius $2$.
The parametric equation of the circle is $-2cos\theta+2$ and $2sin\theta$
$D^2=(u-2)^2+v^2$
Also $D^2=(u+2cos\theta-2)^2+(v-2sin\theta)^2$
How we will proceed from here
 [/INST] The distance between the circle and $P(u,v)$ is the distance between $P$ and the centre minus the radius. So we have$$\left|\sqrt{(u+2)^2+v^2}-2\right|=\sqrt{(u-2)^2+v^2}$$Squaring,$$\begin{align*}&(u+2)^2+v^2+4-4\sqrt{(u+2)^2+v^2}=(u-2)^2+v^2\\\implies&2u+1=\sqrt{(u+2)^2+v^2}\\\implies&4u^2+4u+1=(u+2)^2+v^2\\\implies&u^2-v^2/3=1.\end{align*}$$
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