Question

The circle $x^2+y^2-2x+2y-14=0$ cuts chords of length $2\sqrt{3}$ from lines which belong to the family of lines $(x+5y-22)-\lambda (x-8y+30)=0$. Determine the equations of the lines of the family.

Answer 1

$2\sqrt{r^2-p^2}=2\sqrt{3}\Rightarrow r^2-p^2=3$ .........(1)
Circle is (1,-1) , r = 4 $\therefore p^2=r^2-3=13$ ....(2)
Line is $(1-\lambda )x+(5+8\lambda )y-(22+30\lambda )=0$
$\therefore p=\frac{(1-\lambda )\cdot 1-(5+8\lambda )-(22+30\lambda )}{\sqrt{\left[\right(1-\lambda {)}^2+(5+8\lambda {)}^2]}}$
Squaring and simplifying, we get
$(5{\lambda }^2+6\lambda +2)\cdot {13}^2={13}^2(2+3\lambda {)}^2$
or $2{\lambda }^2+3\lambda +1=0\therefore \lambda =-1,-1/2$
Ans.$2x-3y+8=0,3x+2y-14=0$