Question

If points (0,0) (2,0) (0,-2) (K,-2) are concyclic then k=

Answer 1

Assume a square ABCD

This is a cyclic quadrilateral.
In Line segment AC,
Let O(x, y) be the midpoint.
Then, by section formula,
Then let the ratio between AO:OC=1:1
Coordinates of O is =(nx1+mx2/m+n,ny1+my2/m+n)
=>(x,y)=(1*0+1*0/1+1,1*0+1*-2/1+1)
=>(x,y)=(0+0/2,0-2/2)
=>(x,y)=(0,-1)
Therefore Coordinates of O is (0,-1)
Now,in line segment BD,O is the also the midpoint.
Then BO:OD=1:1.
Then Coordinates of O=( nx1+mx2/m+n, ny1+my2/m+n)
=>(0,-1)=(1*2+1*K/1+1,1*0+1*-2/1+1)
=>(0,-1)=(2+K/2,-1)
Now,by substituting the value 0 = 2+K/2,
We get an equation, 0=2+K
Therefore K= -2(ANSWER).

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