The circle c is enclosed in the square PQRS and touches all four sides, as shown in the diagram - Leaving Cert Mathematics - Question 4 - 2019

The centre of circle c can be found by calculating the midpoint of the line segment connecting the opposite vertices of the square. In this case, we can take vertices P and R for calculation.

Using the midpoint formula:

o = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\n o = \left( \frac{2 + 4}{2}, \frac{3 + 17}{2} \right) = (3, 10)

Therefore, the co-ordinates of the centre c are (3, 10).

To find the radius of circle c, we can calculate the distance from the center (3, 10) to any of the square's sides. We can choose to calculate the distance from this point to point P (2, 3):

Using the distance formula:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 3)^2 + (3 - 10)^2} = \sqrt{1 + 49} = \sqrt{50}

Thus, the radius of c is 5 units.

The standard form of the equation of a circle is:

$\left( x - h \right)^2 + \left( y - k \right)^2 = r^2$

where ( (h, k) ) are the co-ordinates of the center, and ( r ) is the radius. We have

• Centre: (3, 10)
• Radius: 5

Substituting these values into the formula gives:

$\left( x - 3 \right)^2 + \left( y - 10 \right)^2 = 5^2$

Therefore, the equation of circle c is:

$\left( x - 3 \right)^2 + \left( y - 10 \right)^2 = 25$.