Explanation
Step 1: Common Root α ki value nikalna
Kyunki α dono equations ka root hai:
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14α2−31α+3λ=0 --- (Eq. 1)
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35α2−53α+4λ=0 --- (Eq. 2)
λ ko khatam karne ke liye, (Eq. 1) ko 4 se aur (Eq. 2) ko 3 se multiply karke subtract karte hain:
(56α2−124α+12λ)−(105α2−159α+12λ)=0
−49α2+35α=0
−7α(7α−5)=0
Sawaal mein diya hai λ=0, isliye α=0 hoga. Toh:
α=75
Step 2: β aur γ ki values λ ke terms mein
Roots ke product ka formula (P=ac) istemal karte hain:
Ab humein jin roots ki equation nikalni hai, unhe simplify karte hain:
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First root (x1): β3α=(3λ/14α)3α=λ14α2
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Second root (x2): γ4α=(4λ/35α)4α=λ35α2
Step 3: λ ki value nikalna
α=75 ko (Eq. 1) mein rakhte hain:
14(75)2−31(75)+3λ=0
14(4925)−7155+3λ=0
750−7155+3λ=0⟹−7105+3λ=0⟹−15+3λ=0
λ=5
Step 4: Naye roots ki numerical value
Step 5: Quadratic Equation banana
Equation ka formula: x2−(x1+x2)x+(x1x2)=0
Sum of roots: 710+725=735=5
Product of roots: 710×725=49250
Equation hogi:
x2−5x+49250=0
49x2−245x+250=0
Sahi vikalp (Correct option) hai: (1) 49x2−245x+250=0
