The Circle (Leaving Cert Mathematics): Revision Notes

Worked Example 1: Circle with centre at origin

Find the coordinates of the points where: (i) the circle $x^2 + y^2 = 16$ intersects the x-axis (ii) the circle $(x + 3)^2 + (y - 2)^2 = 10$ intersects the y-axis

Solution:

(i) Finding x-axis intersections for $x^2 + y^2 = 16$

Since the circle intersects the x-axis where y = 0:

Substitute y = 0: $x^2 + 0^2 = 16$ $x^2 = 16$ $x = ±4$

Therefore, $x^2 + y^2 = 16$ intersects the x-axis at (4, 0) and (-4, 0).

(ii) Finding y-axis intersections for $(x + 3)^2 + (y - 2)^2 = 10$

Since the circle intersects the y-axis where x = 0:

Substitute x = 0: $(0 + 3)^2 + (y - 2)^2 = 10$ $9 + (y - 2)^2 = 10$ $(y - 2)^2 = 1$ $y - 2 = ±1$ $y = 2 ± 1$ $y = 3 \text{ or } y = 1$

Therefore, $(x + 3)^2 + (y - 2)^2 = 10$ intersects the y-axis at (0, 1) and (0, 3).

Worked Example 2: Circle touching the axes

Consider a circle with centre C and radius length 3 that touches both the x-axis and y-axis. When a circle touches an axis (is tangent to it), it intersects at exactly one point.

If a circle with centre (a, b) and radius r is tangent to:

• The x-axis: the distance from centre to x-axis equals the radius, so |b| = r
• The y-axis: the distance from centre to y-axis equals the radius, so |a| = r