Home
Class 12
MATHS
If the sum of the roots of the quadratic...

If the sum of the roots of the quadratic equation `a x^2+b x+c=0` is equl to the sum of the squares of their reciprocals, then prove that `a/c , b/a a n d c/b` are in H.P.

Text Solution

Verified by Experts

Let `alpha,beta` be roots of `ax^(2)+bx+c=0`. Then
`alpha+beta=1/(alpha^(2))+1/(beta^(2))=(beta^(2)+alpha^(2))/(alpha^(2)beta^(2))`
`rArralpha+beta=((alpha+beta)^(2)-2alphabeta)/(alpha^(2)beta^(2))`
`rArr(-b)/a=((-b//a)^(2)-2c/a)/((c//a)^(2))`
or `(-b)/a=(b^(2)-2ac)/c^(2)`
or `-bc^(2)=ab^(2)-2a^(2)c`
or `2a^(2)c=ab^(2)+bc^(2)`
or `(2a)/b=b/c+c/a`
`rArrc/a,a/b,b/c` are in A.P.
`rArra,c,b/a,c/b` are in H.P.
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERICISE 5.7|4 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERICISE 5.8|10 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERICISE 5.5|10 Videos

  • PROBABILITY II

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

Similar Questions

Explore conceptually related problems

If the sum of the roots of the quadratic equation a x^2+b x+c=0 is equal to the sum of the squares of their reciprocals, then prove that a/c , b/a a n d c/b are in H.P.

If the sum of the roots of the quadratic equation ax^(2)+bx+c=0 be equal to the sum of their squares, then prove that 2ac=ab+b^(2) .

If the sum of the roots of the equation 2x^(2)+4x+C=0 be equal to the sum of their square then

Show that the sum of roots of a quadratic equation ax^(2)+bx+c=0 (a ne 0) is (-b)/(a) .

If the ratio of the roots of the equation x^2+p x+q=0 are equal to ratio of the roots of the equation x^2+b x+c=0 , then prove that p^2c=b^2qdot

If the roots of the quadratic equation (b-c)x^2+(c-a)x+(a-b)= 0 are equal, then prove that 2b=a+c.

If the ratio of the roots of the quadratic equation ax^(2)+bx+c=0 be m:n, then prove that ((m+n)^2)/(mn)=(b^(2))/(ac)

If the roots of the quadratic equation a(b-c)x^2+b(c-a)+c(a-b)=0 be equal then prove that a,b,c are in Harmonic progression .

If the roots of the quadratic equation (b-c)x^(2)+(c-a)x+(a-b)=0 are real and equal then prove that 2b=a+c.

If the roots of the quadratic equation (a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0 are equal, prove that a/b=c/d .