Given:

The centre of the circle is given as (h, k) = (2,2)

We know that the circle passes through point (4,5), the radius (r) of the circle is the distance between the points (2,2) and (4,5).

r = \( \sqrt{(2-4)^2 + (2-5)^2}\)

= \( \sqrt{(-2)^2 + (-3)^2}\)

= \( \sqrt{4+9}\)

= \( \sqrt{13}\)

The equation of the circle is given as

(x– h)^{2}+ (y – k)^{2} = r^{2}

(x –h)^{2} + (y – k)^{2} = \( ( \sqrt{13})^2\)

(x –2)^{2} + (y – 2)^{2} = \( ( \sqrt{13})^2\)

x^{2} – 4x + 4 + y^{2 }– 4y + 4 = 13

x^{2} + y^{2} – 4x – 4y = 5

The equation of the required circle is x^{2} + y^{2} – 4x – 4y = 5

An equilateral triangle is inscribed in the parabola y^{2} = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

Find the area of the triangle formed by the lines joining the vertex of the parabola x^{2} = 12y to the ends of its latus rectum.

A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.