Question

Let S(3, 4) and S(9, 12) be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus S to a tangent to the ellipse is (1, –4), then the eccentricity of the ellipse is 

Solution

Correct option is

5/13

 

Let e be the eccentricity of the ellipse. Also, let the lengths of semi-major and minor axes be a and b respectively. 

Then, 

         SS’ = 2ae

  

Since centre of the ellipse is the mid-point of SS’. So, the coordinates of the centre are 

The equation of the auxiliary circle is  

          

We know that the foot of the perpendicular from the focus on any tangent lies on the auxiliary circle. Therefore (1, –4) lies on the auxiliary circle (ii). 

        

Now, ae = 5 and a = 13  

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SIMILAR QUESTIONS

Q1

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.

Q2

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

Q3

 

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

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

Q4

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

Q5

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 

Q6

 

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

Q7

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 

Q8

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

Q9

If α – β = constant, then the locus of the point of intersection of tangents at  to the ellipse   

Q10

Let S and S’ be two foci of the ellipse . If a circle described on SS’ as diameter intersects the ellipse in real and distinct points, then the eccentricity e of the ellipse satisfies