Home
Class 12
MATHS
If normal to hyperbola x^2/a^2-y^2/b^2=1...

If normal to hyperbola `x^2/a^2-y^2/b^2=1` drawn at an extremity of its latus-rectum has slope equal to the slope of line which meets hyperbola only once, then the eccentricity of hyperbola is

A

`e = sqrt((1+sqrt(5))/(2))`

B

`e = sqrt((sqrt(5)+3)/(2))`

C

`e = sqrt((2)/(sqrt(5)-1))`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
Given hyperbola is `(x^(2))/(a^(2))-(y^(2))/(b^(2)) =1`
Differentiate w.r.t.x, we get` (2x)/(a^(2))-(2yy')/(b^(2)) =0` or `y' =(b^(2)x)/(a^(2)y)`
Slope of normal at `(ae,(b^(2))/(a))` is `y' = - (a^(2)(b^(2))/(a))/(b^(2)ae) =- (1)/(e)`
Now line which meets hyperbola only once has slope `+- (b)/(a)` (parallel to asymptote)
According to question `-(b)/(a) =- (1)/(e)`
`rArr (b^(2))/(a^(2)) = (1)/(e^(2))`
`rArr e^(2) -1 =(1)/(e^(2))`
`rArr e^(4) -e^(2) -1 =0`
`rArr e^(2) = (1+- sqrt(5))/(2)`
`rArr e = sqrt((1+sqrt(5))/(2)`
Promotional Banner

Similar Questions

Explore conceptually related problems

The eccentricity of the hyperbola x^2-y^2=4 is

The line 2x + y = 1 is tangent to the hyperbola x^2/a^2-y^2/b^2=1 . If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

The line 2x + y = 1 is tangent to the hyperbola x^2/a^2-y^2/b^2=1 . If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

If P Q is a double ordinate of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 such that O P Q is an equilateral triangle, O being the center of the hyperbola, then find the range of the eccentricity e of the hyperbola.

PQ is the double ordinate of the hyperbola x^2/a^2-y^2/b^2=1 and O is the centre of the hyperbola. If triangle OPQ is an equilateral triangle, then show that the eccentricity of this hyperbola is e) e^2 >4/3

If the latus rectum of a hyperbola is equal to half of its transverse axis, then its eccentricity is -

The tangent at a point P on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 passes through the point (0,-b) and the normal at P passes through the point (2asqrt(2),0) . Then the eccentricity of the hyperbola is

Let L L ' be the latus rectum through the focus of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 and A ' be the farther vertex. If A ' L L ' is equilateral, then the eccentricity of the hyperbola is (axes are coordinate axes).

If the normal at a point P to the hyperbola x^2/a^2 - y^2/b^2 =1 meets the x-axis at G , show that the SG = eSP.S being the focus of the hyperbola.

The eccentricity of the hyperbola 4x^(2)-9y^(2) =36 is: