The locus of the foot of the perpendicular drawn from a focus onto a tangent to the hyperbola

Question

Passage: The locus of the foot of the perpendicular drawn from a focus onto a tangent to the hyperbola is the auxiliary circle. Consider the hyperbola whose foci are (5, 6) and ( − 3, − 2). The foot of the perpendicular from the focus upon a tangent is the point (2, 5).

Answer the following questions.

 (i)  Length of the conjugate axis of the hyperbola is

(a)   4√22

(b)   2√22

(c)   2√11

(d)   4√11

(ii) The directrix of the hyperbola corresponding to the focus (5, 6) is

(a)  2x + 3y - 11 = 0

(b)  2x + 2y - 1 = 0

(c)  2x + 3y - 9 = 0

(d)  2x + 2y - 7 = 0

(iii)  Length of the latus rectum of the hyperbola

(a)  44/√10

(b)  22/√10

(c)   32/√10

(d)   42/√10

Answer 1

Correct option  (i) (b)(ii) (a)(iii)(a)

Explanation :

(i) The centre of the hyperbola is

(-3 + 5/2,6- 2/2) = (1,2)

Since (2, 5) lies on the auxiliary circle, its radius is

The distance between the foci is 2ae. From the coordinates of the foci, the distance between them is

 (ii) Let the corresponding directrix be x + y = λ (because the directrix is perpendicular to transverse axis). The distance of this from the centre is a/e

Therefore, the directrix equation is 2x + 2y − 11 = 0.

 (iii) The length of the latus rectum is