Home
Class 12
MATHS
The line 2x + y = 1 is tangent to the h...

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

Text Solution

Verified by Experts

The correct Answer is:
B
Substituting `(a//e,0)` in `y=-2x+1`, we get
`0=-(2a)/(e)+1`
`"or "(2a)/(e)=1`
`"or "a=(e)/(2)`
Also, `1=sqrt(a^(2)m^(2)-b^(2))`
`"or "1=4a^(2)-b^(2)`
`"or "1=(4e^(2))/(4)-b^(2)`
`"or "b^(2)=e^(2)-1`
Also, `b^(2)=a^(2)(e^(2)-1)`
`therefore" "a=1,e=2`
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise JEE MAIN|3 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise JEE ADVANCED|6 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise MATRIX MATHC TYPE|10 Videos

  • HIGHT AND DISTANCE

    CENGAGE PUBLICATION|Exercise Archives|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer Type|2 Videos

Similar Questions

Explore conceptually related problems

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

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x-axis. Then the eccentricity of the hyperbola 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 P(6,3) be a point on the hyperbola parabola x^2/a^2-y^2/b^2=1 If the normal at the point intersects the x-axis at (9,0), then the eccentricity of the hyperbola is

If the straight line y =kx+3 is a tangent to the hyperbola 7x^2-4y^2=28 , find K

If the line 2x+sqrt(6)y=2 is a tangent to the hyperbola x^(2)-2y^(2)=4 , then the coordinates of the point of contact are -

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

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

Let P(4,3) be a point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 . If the normal at P intersects the x-axis at (16,0), then the eccentriclty of the hyperbola is

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 the range of eccentricity