Home
Class 12
MATHS
The number of normal (s) of a rectangula...

The number of normal (s) of a rectangular hyperbola which can touch its conjugate is equal to

A

0

B

2

C

4

D

8

Text Solution

Verified by Experts

The correct Answer is:
C
Normal to hyperbola `xy = c^(2)` at `(ct, (c )/(t))` is `y - (c )/(t) = t^(2) (x-ct)` Solving with `xy =- c^(2)`, we get
`rArr x {(c )/(t) + t^(2) (x-ct)} +c^(2) =0`
`rArr t^(2) x^(2) + ((c )/(t)-ct^(3)) x +c^(2) =0`
Since line touches the curve, above equation has equal roots
`:. Delta = 0, ((1)/(t)-t^(3))^(2) - 4t^(2) =0`
`rArr (1-t^(4))^(2) - 4t^(4) =0`
`rArr t^(2) = (2=- sqrt(8))/(2)` or `(-2 +- sqrt(8))/(2)`
`rArr t^(2) =1 + sqrt(2)` or `-1 + sqrt(2)`
Thus four such values of t are possible
Promotional Banner

Similar Questions

Explore conceptually related problems

The vertices of Delta ABC lie on a rectangular hyperbola such that the orthocenter of the triangle is (3, 2) and the asymptotes of the rectangular hyperbola are parallel to the coordinate axes. The two perpendicular tangents of the hyperbola intersect at the point (1, 1). The number of real tangents that can be drawn from the point (1, 1) to the rectangular hyperbola is

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

If P N is the perpendicular from a point on a rectangular hyperbola x y=c^2 to its asymptotes, then find the locus of the midpoint of P N

Four points are such that the line joining any two points is perpendicular to the line joining other two points. If three points out of these lie on a rectangular hyperbola, then the fourth point will lie on the saem hyperbola the conjugate hyperbola one of the directrix one of the asymptotes

The distance between two directrices of a rectangular hyperbola is 10 units. Find the distance between its foci.

If P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3) and S(x_4,y_4) are four concyclic points on the rectangular hyperbola and xy = c^2 , then coordinates of the orthocentre ofthe triangle PQR is

P Q and R S are two perpendicular chords of the rectangular hyperbola x y=c^2dot If C is the center of the rectangular hyperbola, then find the value of product of the slopes of C P ,C Q ,C R , and C Sdot

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is : (1) 4/3 (2) 4/(sqrt(3)) (3) 2/(sqrt(3)) (4) sqrt(3)