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If the roots of the equation x2 + px + q = 0 are in the same ratio as those of the equation x2 + lx + m = 0, then which one of the following is correct?
- a)p2m = l2q
- b)m2p = l2q
- c)m2p = q2l
- d)m2p2 = l2q
Correct answer is option 'A'. Can you explain this answer?
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If the roots of the equation x2 + px+ q= 0 are in the same ratio as th...
Let α, kα be roots of x2 + px + q = 0 and β , kβ be roots of x2 + ℓx + m = 0 Clearly,
Most Upvoted Answer
If the roots of the equation x2 + px+ q= 0 are in the same ratio as th...
Given equations:
1) x^2 + px + q = 0
2) x^2 + lx + m = 0
Let the roots of equation 1 be α and β, and the roots of equation 2 be ξ and η.
Ratio of roots:
The roots of a quadratic equation ax^2 + bx + c = 0 are given by the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
From the above equation, we can see that if x1 and x2 are the roots of a quadratic equation, then the ratio of the roots is given by:
x1/x2 = (-b + √(b^2 - 4ac))/(-b - √(b^2 - 4ac))
= (√(b^2 - 4ac) - b)/(-√(b^2 - 4ac) - b)
= - (√(b^2 - 4ac) - b)/(√(b^2 - 4ac) + b)
Let's find the roots of equations 1 and 2 and compare their ratios.
Roots of equation 1:
Using the quadratic formula, we have:
α = (-p + √(p^2 - 4q))/2
β = (-p - √(p^2 - 4q))/2
Ratio of roots of equation 1:
α/β = (-p + √(p^2 - 4q))/(-p - √(p^2 - 4q))
= (√(p^2 - 4q) - p)/(-√(p^2 - 4q) - p)
= - (√(p^2 - 4q) - p)/(√(p^2 - 4q) + p)
Roots of equation 2:
Using the quadratic formula, we have:
ξ = (-l + √(l^2 - 4m))/2
η = (-l - √(l^2 - 4m))/2
Ratio of roots of equation 2:
ξ/η = (-l + √(l^2 - 4m))/(-l - √(l^2 - 4m))
= (√(l^2 - 4m) - l)/(-√(l^2 - 4m) - l)
= - (√(l^2 - 4m) - l)/(√(l^2 - 4m) + l)
Given that the ratios of the roots are the same, we can equate α/β and ξ/η:
- (√(p^2 - 4q) - p)/(√(p^2 - 4q) + p) = - (√(l^2 - 4m) - l)/(√(l^2 - 4m) + l)
Simplifying the equation:
(√(p^2 - 4q) - p)/(√(p^2 - 4q) + p) = (√(l^2 - 4m) - l)/(√(l^2 - 4m) + l)
1) x^2 + px + q = 0
2) x^2 + lx + m = 0
Let the roots of equation 1 be α and β, and the roots of equation 2 be ξ and η.
Ratio of roots:
The roots of a quadratic equation ax^2 + bx + c = 0 are given by the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
From the above equation, we can see that if x1 and x2 are the roots of a quadratic equation, then the ratio of the roots is given by:
x1/x2 = (-b + √(b^2 - 4ac))/(-b - √(b^2 - 4ac))
= (√(b^2 - 4ac) - b)/(-√(b^2 - 4ac) - b)
= - (√(b^2 - 4ac) - b)/(√(b^2 - 4ac) + b)
Let's find the roots of equations 1 and 2 and compare their ratios.
Roots of equation 1:
Using the quadratic formula, we have:
α = (-p + √(p^2 - 4q))/2
β = (-p - √(p^2 - 4q))/2
Ratio of roots of equation 1:
α/β = (-p + √(p^2 - 4q))/(-p - √(p^2 - 4q))
= (√(p^2 - 4q) - p)/(-√(p^2 - 4q) - p)
= - (√(p^2 - 4q) - p)/(√(p^2 - 4q) + p)
Roots of equation 2:
Using the quadratic formula, we have:
ξ = (-l + √(l^2 - 4m))/2
η = (-l - √(l^2 - 4m))/2
Ratio of roots of equation 2:
ξ/η = (-l + √(l^2 - 4m))/(-l - √(l^2 - 4m))
= (√(l^2 - 4m) - l)/(-√(l^2 - 4m) - l)
= - (√(l^2 - 4m) - l)/(√(l^2 - 4m) + l)
Given that the ratios of the roots are the same, we can equate α/β and ξ/η:
- (√(p^2 - 4q) - p)/(√(p^2 - 4q) + p) = - (√(l^2 - 4m) - l)/(√(l^2 - 4m) + l)
Simplifying the equation:
(√(p^2 - 4q) - p)/(√(p^2 - 4q) + p) = (√(l^2 - 4m) - l)/(√(l^2 - 4m) + l)
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If the roots of the equation x2 + px+ q= 0 are in the same ratio as those of the equation x2 + lx+ m= 0, then which one of the following is correct?a)p2m= l2qb)m2p= l2qc)m2p= q2ld)m2p2 = l2qCorrect answer is option 'A'. Can you explain this answer?
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