Ellipses which are drawn with the same two perpendicular lines as axes and with the sum of the reciprocals of squares of the lengths of their semi-major axis and semi-minor axis equal to a constant have only

Options

A. two points in common
B. four points in common
C. six points in common
D. eight points in common

Answer

B

Explanation

Let the two perpendicular lines be the coordinate axes and origin be the centre of the ellipse.
Let the equation of the ellipse be
`x^(2)/a^(2) + y^(2)/b^(2) = 1`
It is given that `1/a^(2) + 1/b^(2) = 1/k^(2)` (a constant). So, the equation of the ellipse becomes
`1/a^(2) (x^(2) - y^(2)) + y^(2)/k^(2) - 1 = 0`
The represents a family of curves passing through the intersection of `x^(2) - y^(2) = 0 and y^(2)/k^(2) - 1 = 0`
i.e. the points `(pm k, pm) or, (k, k), (-k, -k), (k, -k) and (-k, k).`
Hence, every member of the family passes through the four points.