Centre (-a, -b) and radius \( \sqrt{a^2-b^2}\)

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) = (-a, -b) and radius (r) = \( \sqrt{a^2-b^2}\)

The equation of the circle is

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

x^{2} + 2ax + a^{2} + y^{2} + 2by + b^{2} = a^{2} – b^{2}

x^{2} + y^{2} +2ax + 2by + 2b^{2} = 0

The equation of the circle is x^{2} + y^{2} +2ax + 2by + 2b^{2} = 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.