Question

Find the equation to the circle touching the y-axis at a distance – 3 from the origin and intercepting a length 8 on the x-axis.

Answer 1

Let the equation of the circle be x² + y² + 2gx + 2fy + c = 0. 

Since it touches y-axis at (0, –3) and (0, – 3) lies on the circle. 

∴ c = f² ...(i) 9 – 6f + c = 0 .......(ii) 

From (i) and (ii), we get 9 – 6f + f² = 0 

⇒ (f – 3)² = 0 ⇒ f = 3. 

Putting f = 3 in (i) we obtain c = 9. 

It is given that the circle x² + y² + 2gx + 2fy + c = 0 intercepts length 8 on x-axis

Hence, the required circle is x² + y² ± 10x + 6y + 9 = 0.