Question

If the points (2 , 0) , (0 , 1) , (4 , 5) and (0 , C) are concyclic , then the value of C is

• A. $1$
• B. $\frac{14}{3}$
• C. $5$
• D. None of these

Answer 1

The correct option is A $1$

first, we need to find the equation of circle passes through the point$(2,0),(0,1),(4,5)$

equation of the circle

$x^2+y^2+2gx+2fy+c=0$

so put the points

$4+4g+c=0$.............1

$1+2f+c=0$..............2

$39+8g+10f+c=0$............3

solve these equation

$g=-\frac{25}{12}$

$f=-\frac{8}{3}$

$c=\frac{13}{3}$

so equation of circle

$x^2+y^2-\frac{25x}{6}-\frac{16y}{3}+\frac{13}{3}=0$

now point$(0,c)$ satisfy this equation

$c^2-\frac{16c}{3}+\frac{13}{3}=0$

$3c^2-16c+13=0$

$c=1,\frac{13}{3}$