Home
Class 12
MATHS
If the parabolas y^(2)=4b(x-c) " and" y^...

If the parabolas `y^(2)=4b(x-c) " and" y^(2)=8ax` have a common normal other than x-axis, then which one of the following is a valid choice for the ordered traid (a, b, c) ?

A

`((1)/(2), 2,0)`

B

(1,1,0)

C

(1,1,3)

D

`((1)/(2),2,3)`

Text Solution

Verified by Experts

The correct Answer is:
C
Normal to parabola `y^(2) = 4 ax` is given by
`y = mx - 2 am - am^(3)`
`:.` Normal to parabola
`y^(2) = 4b (x - c)` is
`y = m(x - c) - 2bm - bm^(3)`
[replacing a by b and x by x - c]
`= mx - (2b + c) m = bm^(3)`

and normal ot parabola `y^(2) = 8` ax is
`y = mx - 4a - 2am^(3)`
[replacing a by 2a]
For common normal, we should have
`mx - 4am - 2am^(3) = mx - (2b + c) m - bm^(3)`
[using Eqs. (i) and (ii)]
`4 am + 3am^(2) = (2b + c) m + bm^(3)`
`implies (2a - b) m^(3) + (4a - 2b - c) m = 0`
`implies m((2a - b) m^(2) + (4a - 2b - c)) = 0`
`implies m = 0`
or `m^(2) = (2b + c+ 4a)/(2a - b) = (c )/(2a - b) - 2`
As, `m^(2) gt 0`, therefore `(C )/(2a - b) gt 2`
Note that if m = 0 then all options satisfy
(`:' y = 0` is common normal) and if common normal is other than the axis, then only option (c ) satisfies
[`:'` for option (c ), `2a - b = (3)/(2 - 1) = 3 gt 2`]
Promotional Banner

Similar Questions

Explore conceptually related problems

If the parabolas y^2=4a x and y^2=4c(x-b) have a common normal other than the x-axis (a , b , c being distinct positive real numbers), then prove that b/(a-c)> 2.

If ax^(2)+ bx +c and bx ^(2) + ax + c have a common factor x +1 then show that c=0 and a =b.

If y^2=4b(x-c) and y^2 =8ax having common normal then (a,b,c) is (a) (1/2,2,0) (b) (1,1,3) (c) (1,1,1) (d) (1,3,2)

The axis of the parabola x^(2)=-4y is ……. .

A square has one vertex at the vertex of the parabola y^2=4a x and the diagonal through the vertex lies along the axis of the parabola. If the ends of the other diagonal lie on the parabola, the coordinates of the vertices of the square are (a) (4a ,4a) (b) (4a ,-4a) (c) (0,0) (d) (8a ,0)

If the normal chhord of the parabola y^(2)=4x makes an angle 45^(@) with the axis of the parabola, then its length, is

If a normal to a parabola y^2 =4ax makes an angle phi with its axis, then it will cut the curve again at an angle

If the parabola y=(a-b)x^2+(b-c)x+(c-a) touches x- axis then the line ax+by+c=0 passes through a fixed point

There si a parabola having axis as x -axis, vertex is at a distance of 2 unit from origin & focus is at (4,0) . Which of the following point does not lie on the parabola. (a) (6,8) (b) (5,2sqrt(6)) (c) (8,4sqrt(3)) (d) (4,-4)