Question

Find the equations of the circles touching y-axis at (0, 3) and making an intercept of 8 units on the X-axis.

Answer 1

Case I: The centre lies in first quadrant.


Let the required equation be ${\left(x-h\right)}^2+{\left(y-k\right)}^2=a^2$.

Here, AB = 8 units and L (0, 3)

In $△$CAM:
$\Rightarrow C{A}^2=C{M}^2+A{M}^2$
$$\begin{gathered} \Rightarrow C{A}^2=3^2+4^2 \\ \Rightarrow CA=5 \\ \Rightarrow CL=CA=5 \end{gathered}$$

∴ Coordinates of the centre = $\left(5,3\right)$
And, radius of the circle = 5
${\left(x-5\right)}^2+{\left(y-3\right)}^2=25$, i.e. $x^2+y^2-10x-6y=-9$


Case II: The centre lies in the second quadrant.


Coordinates of the centre = $\left(-5,3\right)$
And, radius of the circle= 5
${\left(x+5\right)}^2+{\left(y-3\right)}^2=25$, i.e. $x^2+y^2+10x-6y=-9$

Hence, the equation of the required circle is ${\left(x\pm 5\right)}^2+{\left(y-3\right)}^2=25$, i.e. $x^2+y^2\pm 10x-6y=-9$.