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

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.`

Text Solution

Verified by Experts

which parabola `y^(2)=4ax`, equation of normal having slope m is
`y=mx-2am-am^(3)` (1)
For parabola `y^(2)=4c(x-b)`, equation of normal having slope m is `y=m(x-b)-2cm-cm^(3)` (2)
For common normal to parabolas, equations (1) and (2) are identical.
`:." "-2am-am^(3)=-bm-2cm-cm^(3)`
`rArr" "(c-a)m^(2)=2(a-c)-b" (as m" !=0)`
`rArr" "m^(2)=(2(a-c)-b)/(c-a)`
For real value of m,
`(2(a-c)-b)/(c-a)gt0`
`rArr" "-2+(b)/(a-c)gt0`
`rArr" "(b)/(a-c)gt2`
Promotional Banner

Topper's Solved these Questions

  • PARABOLA

    CENGAGE|Exercise Exercise 5.7|9 Videos

  • PARABOLA

    CENGAGE|Exercise Exercise (Single)|98 Videos

  • PARABOLA

    CENGAGE|Exercise Exercise 5.5|9 Videos

  • PAIR OF STRAIGHT LINES

    CENGAGE|Exercise Exercise (Numerical)|5 Videos

  • PERMUTATION AND COMBINATION

    CENGAGE|Exercise Question Bank|4 Videos

Similar Questions

Explore conceptually related problems

If a , b ,c are three distinct positive real numbers in G.P., then prove that c^2+2a b >3a cdot

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x >3.

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x >3.

If a,b and c are distinct positive real numbers in A.P, then the roots of the equation ax^(2)+2bx+c=0 are

If a x^2+b x+c=0a n db x^2+c x+a=0 have a common root and a, b, and c are nonzero real numbers, then find the value of (a^3+b^3+c^3)//a b c

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)

If c is positive and 2a x^2+3b x+5c=0 does not have aby real roots, then prove that 2a-3b+5b>0.

If the length of a focal chord of the parabola y^2=4a x at a distance b from the vertex is c , then prove that b^2c=4a^3dot

The chord A B of the parabola y^2=4a x cuts the axis of the parabola at Cdot If A-=(a t_1 ^2,2a t_1),B-=(a t_2 ^2,2a t_2) , and A C : A B 1:3, then prove that t_2+2t_1=0 .