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# Find the shortest and largest distance from the point (2,-7) to the circle $x^2+y^2-14x-10y-151=0$.

$\begin{array}{1 1}(a)\;2\;and\;28\\(b)\;4\;and\;20\\(c)\;3\;and\;27\\(d)\;5\;and\;25\end{array}$

Let $S=x^2+y^2-14x-10y-151=0$
Substituting the point $(2,-7)$ in this equation, we get
$S_1=(2)^2+(-7)^2-14(2)-10(-7)-151=-56 < 0$
$\therefore P(2,-7)$ inside the circle
Radius of circle $r=\sqrt{(-7)^2+(-5)^2+151}=15$
Centre of circle (7,5)
CP=$\sqrt{(7-2)+(5+7)^2}=13$
Shortest distance =$PA=r-CP=15-13=2$
Largest distance $PB=r+CP=15+13=28$
Hence (a) is the correct option.
edited Mar 24, 2014