Question

Equation of the circle with centre on y-axis and passing through the points $(1,0),(1,1)$ is:

• A. $x^2+y^2-y-1=0$
• B. $x^2+y^2-x-1=0$
• C. $x^2+y^2-x+1=0$
• D. $x^2+y^2-y+1=0$

Answer 1

The correct option is A $x^2+y^2-y-1=0$

Let the centre of the circle be $(0,k)$.

Then, the equation of the circle is $(x-0{)}^2+(y-k{)}^2=r^2$

Circle passes through the points $(1,0)$ and $(1,1)$.

Thus, $1+k^2=r^2....\left(1\right)$ and

$1+(1-k{)}^2=r^2\Rightarrow 2+k^2-2k=r^2.....(2)$

$Eq.\left(1\right)-Eq.\left(2\right)$ We get

$2k-1=0$

Solving these equations, we get, $k=\frac{1}{2}$ and $r^2=\frac{5}{4}$

Thus, the required equation of the circle is :

$x^2+(y-\frac{1}{2}{)}^2=\frac{5}{4}$ i.e., $x^2+y^2-y-1=0$