Question

Find the locus of the mid-points of the chords of the hyperbola  which subtend a right angle at the origin.

Solution

Correct option is

 

Let (hk) be the mid-point of the chord of the hyperbola. Then its equation is   

            

    

The equation of the lines joining the origin to the points of intersection of the hyperbola and the chord (i) is obtained by making homogeneous hyperbola with the help of (i) 

  

                                                

The lines represented by (ii) will be at right angle if   

     Coefficient of x2 + Coefficient of y2 = 0 

  

  

Hence, the locus of (hk) is  

        .

 

SIMILAR QUESTIONS

Q1

Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1, 2) and eccentricity 

Q2

Find the eccentricity of the hyperbola whose latus rectum is half of its transverse axis.

Q3

 

Find the equation of the hyperbola whose foci are (6, 4) and (–4, 4) and eccentricity is 2.

Q4

Obtain the equation of a hyperbola with coordinate axes as principal axes given that the distances of one of its vertices from the foci are 9 and 1 units.

Q5

The foci of a hyperbola coincide with the foci of the ellipse . Find the equation of the hyperbola if its eccentricity is 2.

Q6

For what value of λ does the line y = 2x + λ touches the hyperbola  

Q7

Find the equation of the tangent to the hyperbola x2 – 4y2 = 36 which is perpendicular to the line x – y + 4 = 0.  

Q8

 

Find the equation and the length of the common tangents to hyperbola 

            

Q9

Find the locus of the foot of perpendicular from the centre upon any normal to the hyperbola .

Q10

Find the locus of the poles of normal chords of the hyperbola