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

Correct option is

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, (kk), (–k, –k), (k, –k) and (–kk). 

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

SIMILAR QUESTIONS

Q1

The line x = at2 meets the ellipse  in the real points iff

Q2

On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y are

Q3

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  

Q4

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     

Q5

If circumcentre of an equilateral triangle inscribed in  with vertices having eccentric angle α, β, γ respectively is (x1y1) then 

Q6

Locus of the middle points of all chords of , which are at a distance of 2 units from the vertex of parabola y2 = –8axis

Q7

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

Q8

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

Q9

 

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

4x2 + 9y2 – 8x – 36y + 4 = 0 such that ∠ACB = π/2 then CA–2 +CB–2 is equal to  

Q10

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