Since the centre is at (0, 0) and the major axis is on the y-axis, the equation of the ellipse will be of the form

\( \dfrac{x^2}{a^2}\) +\( \dfrac{y^2}{b^2}\) = 1.............(1)

Where, a is the semi-major axis

The ellipse passes through points (3, 2) and (1, 6). Hence

\( \dfrac{9}{b^2}+\dfrac{4}{a^2}\)…. (2)

\( \dfrac{1}{b^2}+\dfrac{36}{a^2}\) ..........(3)

On solving equations (2) and (3), we obtain b^{2}= 10 and a^{2}= 40.

Thus, the equation of the ellipse is \( \dfrac{x^2}{10}+ \dfrac{y^2}{40}=1\)

or 4x^{2} + y^{2} = 40

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.