Advertisements
Advertisements
प्रश्न
If the roots of the equation (a2 + b2)x2 − 2 (ac + bd)x + (c2 + d2) = 0 are equal, prove that `a/b=c/d`.
Advertisements
उत्तर
The given quadric equation is (a2 + b2)x2 − 2 (ac + bd)x + (c2 + d2) = 0, and roots are real
Then prove that `a/b=c/d`.
Here,
a = (a2 + b2), b = -2 (ac + bd) and c = (c2 + d2)
As we know that D = b2 - 4ac
Putting the value of a = (a2 + b2), b = -2 (ac + bd) and c = (c2 + d2)
D = b2 - 4ac
= {-2(ac + bd)}2 - 4 x (a2 + b2) x (c2 + d2)
= 4(a2c2 + 2abcd + b2 + d2) - 4(a2c2 + a2d2 + b2c2 + b2d2)
= 4a2c2 + 8abcd + 4b2d2 - 4a2c2 - 4a2d2 - 4b2c2 - 4b2d2
= -4a2d2 - 4b2c2 + 8abcd
= -4(a2d2 + b2c2 - 2abcd)
The given equation will have real roots, if D = 0
-4(a2d2 + b2c2 - 2abcd) = 0
a2d2 + b2c2 - 2abcd = 0
(ad)2 + (bc)2 - 2(ad)(bc) = 0
(ad - bc)2 = 0
Square root both sides we get,
ad - bc = 0
ad = bc
`a/b=c/d`
Hence `a/b=c/d`
APPEARS IN
संबंधित प्रश्न
If `x=2/3` and x =−3 are roots of the quadratic equation ax2 + 7x + b = 0, find the values of a and b.
Find the values of k for which the quadratic equation (3k + 1) x2 + 2(k + 1) x + 1 = 0 has equal roots. Also, find the roots.
Find the values of k for which the roots are real and equal in each of the following equation:
4x2 - 3kx + 1 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
k2x2 - 2(2k - 1)x + 4 = 0
Find the values of k for which the given quadratic equation has real and distinct roots:
kx2 + 2x + 1 = 0
Find the value of the discriminant in the following quadratic equation :
`4 sqrt 3 "x"^2 + 5"x" - 2 sqrt 3 = 0`
Solve the following quadratic equation using formula method only
`"x"^2 - 4 sqrt 15 "x" - 4 = 0`
Find the value(s) of k for which the pair of equations
kx + 2y = 3
3x + 6y = 10 has a unique solution.
Find if x = – 1 is a root of the equation 2x² – 3x + 1 = 0.
Find the value of k for which the following equation has equal roots:
(k − 12)x2 + 2(k − 12)x + 2 = 0.
Determine whether the given values of x is the solution of the given quadratic equation below:
6x2 - x - 2 = 0; x = `(2)/(3), -1`.
Solve for x: (x2 - 5x)2 - 7(x2 - 5x) + 6 = 0; x ∈ R.
Find the discriminant of the following equations and hence find the nature of roots: 16x2 - 40x + 25 = 0
Discuss the nature of the roots of the following quadratic equations : `3x^2 - 2x + (1)/(3)` = 0
Find the value(s) of p for which the equation 2x2 + 3x + p = 0 has real roots.
Find whether the following equation have real roots. If real roots exist, find them.
–2x2 + 3x + 2 = 0
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(x + 1)(x – 2) + x = 0
Let α and β be the roots of the equation, 5x2 + 6x – 2 = 0. If Sn = αn + βn, n = 1, 2, 3, ....., then ______.
If 3 is a root of the quadratic equation x2 – px + 3 = 0, then p is equal to ______.
One root of equation 3x2 – mx + 4 = 0 is 1, the value of m is ______.
