Home
Class 12
MATHS
If m is the slope of a tangent to the hy...

If m is the slope of a tangent to the hyperbola `(x^(2))/(a^(2)-b^(2))-(y^(2))/(a^(3)-b^(3)) =1` where `a gt b gt 1` when

A

`(a+b) m^(2)+ab ge (a+b)^(2)`

B

`(a+b)^(2)m+ab ge (a+b)`

C

`abm^(2)+(a+b)ge (a+b)^(2)`

D

`(a+b)m^(2)+a^(2)b^(2)ge (a+b)^(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
For hyperbola `(x^(2))/(A^(2))-(y^(2))/(B^(2)) =1` equation of tangent having slope m is
`rArr y = mx +- sqrt(A^(2)m^(2)-B^(2))`
where `A^(2)m^(2) -B^(2) ge 0`
`rArr m^(2) ge (B^(2))/(A^(2))`
`rArr m^(2) ge (a^(3)-b^(3))/(a^(2)-b^(2))`
`rArr (a+b)m^(2) ge a^(2) + ab + b^(2)`
`rArr (a+b) m^(2) + ab ge (a+b)^(2)`
Promotional Banner

Similar Questions

Explore conceptually related problems

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)).

If the chords of contact of tangents from two points (-4,2) and (2,1) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 are at right angle, then find then find the eccentricity of the hyperbola.

If it is possible to draw the tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 having slope 2, then find its range of eccentricity.

If the tangent at point P(h, k) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 cuts the circle x^(2)+y^(2)=a^(2) at points Q(x_(1),y_(1)) and R(x_(2),y_(2)) , then the vlaue of (1)/(y_(1))+(1)/(y_(2)) is

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

Find the area of the triangle formed by any tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 with its asymptotes.

If the angle between the asymptotes of hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 id (pi)/(3) , then the eccentnricity of conjugate hyperbola is _________.

Find the slope of a common tangent to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and a concentric circle of radius rdot

Find the equations to the common tangents to the two hyperbolas (x^2)/(a^2)-(y^2)/(b^2)=1 and (y^2)/(a^2)-(x^2)/(b^2)=1

If a x+b y=1 is tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 , then a^2-b^2 is equal to 1/(a^2e^2) (b) a^2e^2 b^2e^2 (d) none of these