## Question

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.

### Solution

Four points in common

Let the two perpendicular lines be the coordinate axes and origin be the centre of the ellipse.

Let the equation of the ellipse be

It is given that (a constant). So, the equation of the ellipse becomes

This represents a family of curves passing through the intersection of

i.e., the points (±*k*, ±*k*) or, (*k*, *k*), (–*k*, –*k*), (*k*, –*k*) and (–*k*, *k*).

Hence, every member of the family passes through the four points.

#### SIMILAR QUESTIONS

The line *x* = *at*^{2} meets the ellipse in the real points iff

On the ellipse 4*x*^{2} + 9*y*^{2} = 1, the points at which the tangents are parallel to the line 8*x* = 9*y* are

Tangent is drawn to the ellipse , then the value of θ such that sum of intercepts on axes made by the tangent is minimum is

If *p* and *p*’ denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length *a*, respectively, on a tangent to the ellipse and *r* denotes the focal distance of the point, then

If circumcentre of an equilateral triangle inscribed in with vertices having eccentric angle α, β, γ respectively is (*x*_{1}, *y*_{1}) then

Locus of the middle points of all chords of , which are at a distance of 2 units from the vertex of parabola *y*^{2} = –8*ax*is

A point on the ellipse at a distance equal to the mean of lengths of the semi-major and semi-minor axis from the centre, is

A tangent to the ellipse is cut by the tangent at the extremities of the major axis at *T* and *T*’. The circle on *TT*’ as diameter passes through the point

If C is the centre and A, B are two points on the conic

4*x*^{2} + 9*y*^{2} – 8*x* – 36*y* + 4 = 0 such that ∠*ACB* = π/2 then *CA*^{–2} +*CB*^{–2} is equal to

The eccentricity of the ellipse with centre at the origin which meets the straight line on the axis of *x* and the straight line on the axis of *y* and whose axes lie along the axes of coordinates is