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The quadratic equations x2 - 6x + a = 0 and x2 - cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then what is the common root?
Let α and β are roots of: x2 - 6x + a = 0 Let α and γ are roots of: x2 - cx + 6 = 0 So, α + β = 6 ........ (i) α.β = a ........ (ii) α + γ = c ........ (iii) α.γ = 6 ........ (iv) β/γ = 4/3 ........ (v) Now using (ii) and (iii) α.β/α.γ = a/6 => β/γ = a/6 => using (v), 4/3 = a/6 => So, a = 8 Now , x2 - 6x + a = 0 x2 - 6x + 8 = 0 x = 2 OR 6 As α.β = 8 So α = 2
Let α and β are roots of: x2 - 6x + a = 0 Let α and γ are roots of: x2 - cx + 6 = 0 So, α + β = 6 ........ (i) α.β = a ........ (ii)
α + γ = c ........ (iii) α.γ = 6 ........ (iv)
β/γ = 4/3 ........ (v)
Now using (ii) and (iii) α.β/α.γ = a/6 => β/γ = a/6 => using (v), 4/3 = a/6 => So, a = 8
Now , x2 - 6x + a = 0 x2 - 6x + 8 = 0 x = 2 OR 6 As α.β = 8 So α = 2
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