Question

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

• A. 1
• B. $\frac{13}{3}$
• C. 6
• D. 8

Answer 1

Answer: (A), (B)

The correct options are
A 1
B $\frac{13}{3}$

First find the equation of circle passes through point$(2,0),(0,1)$ and $(4,5)$

Equation of the circle

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

so putting the points

$4+4g+c=0.............\left[1\right]$

$1+2f+c=0..............\left[2\right]$

$39+8g+10f+c=0............\left[3\right]$

solve these equation

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

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

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

so equation of circle

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

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

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

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

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