Find the equations of the circles passing through two points on y - axis at distances 3 from the origin and having radius 5.
Given that we need to find the equations of the circles passing through the points on y - axis at distances 3 from origin and having radius 5.
Since the points are 3 units away from origin on the y - axis, the points will be (0,3) and (0, - 3).
Let us assume the centre of this circle be (h,k).
We know that the equation of the circle with centre (p,q) and having radius ‘r’ is given by:
⇒ (x - p)2 + (y - q)2 = r2
Now we substitute the corresponding values in the equation:
⇒ (x - h)2 + (y - k)2 = 52
⇒ (x - h)2 + (y - k)2 = 25 ..... (1)
Since circle passes through the point (0,3). We substitute this point in eq(1)
⇒ (0 - h)2 + (3 - k)2 = 25
⇒ h2 + (3 - k)2 = 25 ..... - - (2)
Since circle passes through the point (0, - 3). We substitute this point in eq(1)
⇒ (0 - h)2 + ( - 3 - k)2 = 25
⇒ h2 + (3 + k)2 = 25 ..... - - (3)
On solving (2) and (3), we get
⇒ h = ±4 and k = 0
We have circle with centre (±4,0) and having radius 5 units.
We know that the equation of the circle with centre (p, q) and having radius ‘r’ is given by:
⇒ (x - p)2 + (y - q)2 = r2
Now we substitute the corresponding values in the equation:
⇒ (x±4)2 + (y - 0)2 = 52
⇒ x2±8x + 16 + y2 = 25
⇒ x2 + y2±8x - 9 = 0
∴The equations of the circles is x2 + y2±8x - 9 = 0.