Home
Class 12
MATHS
The area of triangle formed by the tange...

The area of triangle formed by the tangents from the point (3, 2) to the hyperbola `x^2-9y^2=9` and the chord of contact w.r.t. the point (3, 2) is_____________

Text Solution

Verified by Experts

The correct Answer is:
8
The hyperbola is
`x^(2)-9y^(2)=9`
`"or "(x^(2))/(9)-(y^(2))/(1)=1`
The equation of tangent is
`y=mx pm sqrt(a^(2)m^(2)-b^(2))" (1)"`
It passes through (3,2). Therefore,
`2=3mpmsqrt(9m^(2)-1)`
`"or "4+9m^(2)-12m=9m^(2)-1`
Solving, we get the values of m as
`m_(1)=(5)//(12) and m_(2)=oo`
Therefore, equations of tangents are
`5x-12y+9=0" (2)"`
and `x - 3 = 0" "(3)`
Now, the equation of chord of contact w.r.t. point P(3,2) is
T = 0
`"or "3x-18y=9`
`"or "x-6y=3" (3)"`
Solving (2) and (4), we get
`x= -5, y = -(4)/(3)`
Solving (3) and (4), we get
x = 3, y = 0
Now, the vertices of triangle are `(3, 2),(3,0), and (-5,-4//3)`.
Therefore,
`"Area"=(1)/(2)|(3,2,1),(3,0,1),(-5,-4//3,1)|`
`="8. sq. units"`
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE|Exercise JEE Main Previous Year|3 Videos

  • HYPERBOLA

    CENGAGE|Exercise JEE Advanced Previous Year|14 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise (Matrix)|5 Videos

  • HIGHT AND DISTANCE

    CENGAGE|Exercise JEE Previous Year|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Question Bank|25 Videos

Similar Questions

Explore conceptually related problems

The area of the triangle formed by the tangent at the point (a, b) to the circle x^(2)+y^(2)=r^(2) and the coordinate axes, is

The area of the triangle formed by the tangent at the point (a,b) to the circle x^(2)+y^(2)=r^(2) and the co-ordinate axes is:

The area of the triangle formed by two tangents from " (1,1) " to " x^(2)+y^(2)+4x+6y+4=0 " and their chord of contact is :

Find the area of the triangle formed by the tangents from the point (4,3) to the circle x^(2)+y^(2)=9 and the line joining their points of contact.

Find the area of the triangle formed by the tangents drawn from the point (4,6) to the circle x^(2)+y^(2)=25 and their chord of contact. Answer: (405sqrt(3))/(52) sq.units

The area of triangle formed by tangents and the chord of contact from (3,4) to y^(2)=2x is

The equation of the chord of contact of tangents from (1,2) to the hyperbola 3x^(2)-4y^(2)=3 , is

The equation of the chord of contact of tangents drawn from a point (2,-1) to the hyperbola 16x^(2)-9y^(2)=144 , is

For the circle x^(2)+y^(2)=r^(2) , find the value of r for which the area enclosed by the tangents drawn from the point P(6,8) to the circle and the chord of contact and the chord of contact is maximum.