Question

If $(-2,2),(x,8)$ and $(6,y)$ are three concyclic points whose centre is (2, 5) then the possible values of $x$ and $y$ are

• A. $x=6,-2,y=8,2$
• B. $x=2,-2,y=5,2$
• C. $x=3,-1,y=1,2$
• D. $x=2,-1,y=1,4$

Answer 1

The correct option is A $x=6,-2,y=8,2$

Let the given points be$A(-2,2),B(x,8),C(6,y)$ and $O(2,5).$
Here, $OA=OB=OC\Rightarrow O{A}^2=O{B}^2=O{C}^2$ .......... (Radii of same circle)

Now, $O{A}^2=(-2-2{)}^2+(2-5{)}^2=(-4{)}^2+(-3{)}^2=16+9=25$
$O{B}^2=(X-2{)}^2+(8-5{)}^2=(X-2{)}^2+9$
$O{C}^2=(6-2{)}^2+(Y-5{)}^2=4^2+(Y-5{)}^2=16+(Y-5{)}^2$

Since, $O{A}^2=O{B}^2=O{C}^2$, we have
$(x-2{)}^2+9=25$ and $16+(y-5{)}^2=25$
$\Rightarrow (x-2{)}^2=16and(y-5{)}^2=9$
$\Rightarrow x-2=\pm 4$ and $y-5=\pm 3$
$\Rightarrow x=6,-2andy=8,2$

Hence, $x=6,-2andy=8,2$