Let us consider the equation of the required circle be (x – h)^{2}+ (y – k)^{2} =r^{2}

We know that the circle passes through (0, 0),

So, (0 – h)^{2}+ (0 – k)^{2} = r^{2}

h^{2} + k^{2} = r^{2}

Now, The equation of the circle is (x – h)^{2 }+ (y – k)^{2} = h^{2} + k^{2}.

It is given that the circle intercepts a and b on the coordinate axes.

i.e., the circle passes through points (a, 0) and (0, b).

So, (a – h)^{2}+ (0 – k)^{2} =h^{2} +k^{2}……………..(1)

(0 – h)^{2}+ (b– k)^{2} =h^{2} +k^{2}………………(2)

From equation (1), we obtain

a^{2} – 2ah + h^{2} +k^{2} = h^{2} +k^{2}

a^{2} – 2ah = 0

a(a – 2h) =0

a = 0 or (a -2h) = 0

However, a ≠ 0; hence, (a -2h) = 0

h = \( \dfrac{a}{2}\)

From equation (2), we obtain

h^{2} – 2bk + k^{2} + b^{2}= h^{2} +k^{2}

b^{2} – 2bk = 0

b(b– 2k) = 0

b= 0 or (b-2k) =0

However, a ≠ 0; hence, (b -2k) = 0

k =\( \dfrac{b}{2}\)

So, the equation is

(x – \( \dfrac{a}{2}\))^{2} + (y – \( \dfrac{b}{2}\))^{2} = (\( \dfrac{a}{2}\))^{2} + (\( \dfrac{b}{2}\))^{2}

\( (\dfrac{2x-a}{2})^2+(\dfrac{2y-b}{2})^2=\dfrac{a^2+b^2}{4}\)

4x^{2} – 4ax + a^{2} +4y^{2} – 4by + b^{2} = a^{2} + b^{2}

4x^{2} + 4y^{2} -4ax – 4by = 0

4(x^{2} +y^{2} -7x + 5y – 14) = 0

x^{2 }+ y^{2 }– ax – by = 0

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