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Quadratic Equation and Inequalities question

2024 · 4 Apr · Shift 1 · Q46
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  5. /2024 · 4 Apr · Shift 1 · Q46

Quadratic Equation and Inequalities question

2024 · 4 Apr · Shift 1 · Q46

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
If 2 and 6 are the roots of the equation ax2+bx+1=0a x^2+b x+1=0ax2+bx+1=0, then the quadratic equation, whose roots are 12a+b\frac{1}{2 a+b}2a+b1​ and 16a+b\frac{1}{6 a+b}6a+b1​, is :
  1. A
    x2+8x+12=0x^2+8 x+12=0x2+8x+12=0
  2. B
    2x2+11x+12=02 x^2+11 x+12=02x2+11x+12=0
  3. C
    4x2+14x+12=04 x^2+14 x+12=04x2+14x+12=0
  4. D
    x2+10x+16=0x^2+10 x+16=0x2+10x+16=0
View written solutionFree

Correct answer: A

  1. Since 222 and 666 are roots of ax2+bx+1=0,a x^2 + b x + 1 = 0,ax2+bx+1=0, by Vieta’s formulas:

  • Sum of roots: 2+6=8=−ba  ⟹  b=−8a2+6=8=-\frac{b}{a} \implies b=-8a2+6=8=−ab​⟹b=−8a

  • Product of roots: 2⋅6=12=1a  ⟹  a=1122\cdot 6=12=\frac{1}{a} \implies a=\frac{1}{12}2⋅6=12=a1​⟹a=121​

Hence, b=−8⋅112=−23.b=-8\cdot \frac{1}{12}=-\frac{2}{3}.b=−8⋅121​=−32​.

  1. Now compute the new roots: 2a+b=2(112)−23=16−46=−122a+b=2\left(\frac{1}{12}\right)-\frac{2}{3}=\frac{1}{6}-\frac{4}{6}=-\frac{1}{2}2a+b=2(121​)−32​=61​−64​=−21​ So, 12a+b=−2.\frac{1}{2a+b}=-2.2a+b1​=−2.

Similarly, 6a+b=6(112)−23=12−23=−166a+b=6\left(\frac{1}{12}\right)-\frac{2}{3}=\frac{1}{2}-\frac{2}{3}=-\frac{1}{6}6a+b=6(121​)−32​=21​−32​=−61​ So, 16a+b=−6.\frac{1}{6a+b}=-6.6a+b1​=−6.

Thus the required quadratic has roots −2-2−2 and −6-6−6.

  1. Form the quadratic equation with roots −2-2−2 and −6-6−6:
  • Sum of roots: −2+(−6)=−8-2+(-6)=-8−2+(−6)=−8
  • Product of roots: (−2)(−6)=12(-2)(-6)=12(−2)(−6)=12

Therefore, the equation is x2−(sum)x+(product)=0x^2-(\text{sum})x+(\text{product})=0x2−(sum)x+(product)=0 x2−(−8)x+12=0x^2-(-8)x+12=0x2−(−8)x+12=0 x2+8x+12=0.x^2+8x+12=0.x2+8x+12=0.

  1. Compare with options:
  • A: x2+8x+12=0x^2+8x+12=0x2+8x+12=0 ✅
  • B: 2x2+11x+12=02x^2+11x+12=02x2+11x+12=0 ❌
  • C: 4x2+14x+12=04x^2+14x+12=04x2+14x+12=0 ❌
  • D: x2+10x+16=0x^2+10x+16=0x2+10x+16=0 ❌

So the correct option is A.

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