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प्रश्न
If the ratio of the roots of the equation
lx2 + nx + n = 0 is p: q, Prove that
`sqrt(p/q) + sqrt(q/p) + sqrt(n/l) = 0.`
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उत्तर
Let α, β be the roots of
lx2 + nx + n = 0, α + β = `n/l` and αβ = `n/l`.
`α/beta = p/q ...("given")`
Now L.H.S.
= `sqrt(p/q) + sqrt(q/p) + sqrt(n/l)`
= `sqrt(α/beta) + sqrt(beta/α) + sqrt(n/l)`
= `(α + beta)/sqrt(αbeta) + sqrt(n/l)`
= `((-n)/l)/sqrt(n/l) + sqrt(n/l,) [ ∵ α + beta = (-n)/l αbeta = n/l]`
= `-sqrt(n/l) + sqrt(n/l)`
= 0
= R.H.S.
Hence proved.
संबंधित प्रश्न
Determine the nature of the roots of the following quadratic equation:
`3x^2-2sqrt6x+2=0`
Find the values of k for which the roots are real and equal in each of the following equation:
x2 - 2(k + 1)x + (k + 4) = 0
Write the value of k for which the quadratic equation x2 − kx + 4 = 0 has equal roots.
Find the value of the discriminant in the following quadratic equation :
10 x - `1/x` = 3
Solve the following quadratic equation using formula method only :
16x2 = 24x + 1
For what value of k, the roots of the equation x2 + 4x + k = 0 are real?
Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x2 - 4x + 1 = 0
If the roots of ax2 + bx + c = 0 are in the ratio m : n, then:
State whether the following quadratic equation have two distinct real roots. Justify your answer.
x2 – 3x + 4 = 0
Complete the following activity to determine the nature of the roots of the quadratic equation x2 + 2x – 9 = 0 :
Solution :
Compare x2 + 2x – 9 = 0 with ax2 + bx + c = 0
a = 1, b = 2, c = `square`
∴ b2 – 4ac = (2)2 – 4 × `square` × `square`
Δ = 4 + `square` = 40
∴ b2 – 4ac > 0
∴ The roots of the equation are real and unequal.
