Given:

Centre (1, 1) and radius \( \sqrt{2}\)

Let us consider the equation of a circle with centre (h, k) and

Radius r is given as (x – h)^{2 }+ (y – k)^{2 }= r^{2}

So, centre (h, k) = (1, 1) and radius (r) = \( \sqrt{2}\)

The equation of the circle is

(x-1)^{2} + (y-1)^{2 }= \( (\sqrt{2})^2\)

x^{2} – 2x + 1 + y^{2} -2y + 1 = 2

x^{2} + y^{2} – 2x -2y = 0

The equation of the circle is x^{2} + y^{2} – 2x -2y = 0

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.