Home
Class 12
MATHS
A tangent is drawn at any point on the h...

A tangent is drawn at any point on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2)) =1`. If this tangent is intersected by the tangents at the vertices at points P and Q, then which of the following is/are true

A

S,S',P and Q are concyclic

B

PQ is diameter of the circle

C

S,S', P and Q forms rhombus

D

PQ is diagonal of acute angle of the rhombus formed by S,S',P and Q

Text Solution

Verified by Experts

The correct Answer is:
A, B

Any tangent to the hyperbola is `(x sec theta)/(a) -(y tan theta)/(b) =1` Solving this line with the lines `x = +-a`, we get the coordinates of points P and Q as
`(a,b tan.(theta)/(2))` and `(-a,-b cot.(theta)/(2))`
Now slopes of the lines PS and QS are
`m_(PS) =(btan.(theta)/(2))/(a(1-e)), m_(QS) =(-b cot.(theta)/(2))/(-a(1+e))`
`rArr m_(PS).m_(QS) =(-b^(2))/(a^(2)(e^(2)-1)) =-1`
Similarly `m_(PS). m_(QS') =-1`
`rArr` line PQ subtends and angle of `(pi)/(2)` at S and S'
`rArr` points P,Q,S and S' are concyclic.
`rArr PQ` is diameter.
Doubtnut Promotions Banner Mobile Dark
|

Similar Questions

Explore conceptually related problems

The tangent at P on the hyperbola (x^(2))/(a^(2)) -(y^(2))/(b^(2))=1 meets one of the asymptote in Q. Then the locus of the mid-point of PQ is

The tangents drawn from a point P to ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 make angles alpha and beta with the transverse axis of the hyperbola, then

Find the equations of the tangent and normal to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point (x_(0), y_(0)).

Find the equation of pair of tangents drawn from point (4, 3) to the hyperbola (x^(2))/(16)-(y^(2))/(9)=1 . Also, find the angle between the tangents.

Tangents are drawn from any point on the circle x^(2)+y^(2) = 41 to the ellipse (x^(2))/(25)+(y^(2))/(16) =1 then the angle between the two tangents is

Tangents are drawn from any point on the hyperbola (x^2)/9-(y^2)/4=1 to the circle x^2+y^2=9 . Find the locus of the midpoint of the chord of contact.

AB and CD are two equal and parallel chords of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 . Tangents to the ellipse at A and B intersect at P and tangents at C and D at Q. The line PQ

Two tangents are drawn from a point on hyperbola x^(2)-y^(2)=5 to the ellipse (x^(2))/(9)+(y^(2))/(4)=1 . If they make angle alpha and beta with x-axis, then

Tangents are drawn to the hyperbola 4x^2-y^2=36 at the points P and Q. If these tangents intersect at the point T(0,3) then the area (in sq units) of triangle PTQ is

If the tangents to the parabola y^2=4a x intersect the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 at Aa n dB , then find the locus of the point of intersection of the tangents at Aa n dBdot

Home
Profile