Question

 

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

            

Solution

Correct option is

 

  

      

Similarly tangent at any point  on 2nd hyperbolas is  

        

If (i) and (ii) are common tangents then they should be identical. Comparing the coefficients of x and y   

  

    

  

   

  

     

    

Hence the points of contact are  

                                                                                      

Length of common tangent i.e., the distance between the above points is  and equation of common tangent on putting the values of  in (i) is     

       

  

 

Alternative Method : The given two hyperbolas are    

            

  

we know that

 

  

is tangent to (i) for all   

  

         

will be tangent to (ii) 

For common tangents to (i) and (ii) the lines (iii) and (iv) must be identical  

   

     

  

∴ The equation of common tangent lines are  

         

  

equation of tangent to (i) at (x1y1) is 

           

Comparing (v) and (vi), then  

         

  

and equation of tangent to (ii) at (x2y2) is     

            

Comparing (v) and (vii), then  

              

  

Hence the points of contact are  

          

Hence the length of common tangent is 

          .

SIMILAR QUESTIONS

Q1

 

To find the equation of the hyperbola from the definition that hyperbola is the locus of a point which moves such that the difference of its distances from two fixed points is constant with the fixed point as foci.

 

Q2

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

Q3

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

Q4

 

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

Q5

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.

Q6

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

Q7

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

Q8

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

Q9

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

Q10

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