Question

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.