Question

Tangents are drawn to the ellipse  and the circle x2 + y2 = a2 at the points where a common ordinate cuts them (on the same side of the x-axis). Then, the greatest acute angle between these tangents is given by 

Solution

Correct option is

 

Let  be a point on the ellipse  and  be the corresponding point on the auxiliary circle x2 + ya2. The equations of tangents at P and Q to be respective curves are     

        

  

Let α be the acute angle between these tangents. Then, 

      

  

  

  

Hence, the greatest values of   

SIMILAR QUESTIONS

Q1

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

Q2

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

Q3

 

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  

Q4

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.

Q5

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

Q6

 

The radius of the circle passing through the foci of the ellipse

9x2 + 16y2 = 144 and having its centre at (0, 3), is 

Q7

An ellipse has OB as a semi-minor axis, FF’ as its foci and the angle FBF’ is a right angle. Then, the eccentricity of the ellipse is

Q8

The focus of an ellipse is (–1, –1) and the corresponding directix is x – y + 3 = 0. If the eccentricity of the ellipse is 1/2, then the coordinates of the centre of the ellipse are 

Q9

 

The equation of the ellipse with its centre at (1, 2), one focus at (6, 2) and passing through (4, 6) is

Q10

The area of the quadrilateral formed by the tangents at the end-points of latusrecta to the ellipse