Home
Class 12
MATHS
If the normals of the paralbola y^2=4x ...

If the normals of the paralbola y^2=4x drawn at the end points of its latus rectum are tangents to the circle (x-3)^(2) + (y+2)^(2) = r^(2) then the value of r^(2) is

Text Solution

Verified by Experts

The correct Answer is:
2
End points of latusrectum are `(a +- 2a)` i.e. `(1 +- 2)`
Equation of normal at `(x_(1), y_(1))` is
`(y - y_(1))/(x - x_(1)) = (y_(1))/(2a)`
i.e., `(y - 2)/(x - 1) = - (2)/(2)` and `(y + 2)/(x - 1) = (2)/(3)`
`implies x + y + 3`
and `x - y = 3`
Which is tangent to `(X - 3)^(2) + (y + 2)^(2) = r^(2)`

`:.` Length of perpendicular from centre = Radius
`implies (|3 - 2 - 3|)/(sqrt(1^(2) + 1^(2))) = r`
`:. r^(2) = 2`
Promotional Banner

Similar Questions

Explore conceptually related problems

Find the equations of normal to the parabola y^2=4a x at the ends of the latus rectum.

If y = 2sqrt2x+c is a tangent to the circle x^(2) +y^(2) = 16 , find the value of c.

If the normals to the parabola y^2=4a x at the ends of the latus rectum meet the parabola at Q and Q^(prime), then Q Q^(prime) is

Find the area of the parabola y^2=4ax and its latus rectum.

The area of the triangle formed by the tangent and the normal to the parabola y^2=4a x , both drawn at the same end of the latus rectum, and the axis of the parabola is 2sqrt(2)a^2 (b) 2a^2 4a^2 (d) none of these

If the length of the latus rectum rectum of the parabola 169{(x-1)^(2)+(y-3)^(2)}=(5x-12y+17)^(2) is L then the value of 13L/4 is _________.

If the line y = 3x + 1, touches the parabola y^(2) = 4ax , find the length of the latus rectum ?

Equation of a common tangent to the circle x^(2) + y^(2) - 6x = 0 and the parabola, y^(2) = 4x , is