{"@context":"https://schema.org/","@type":"Quiz","name":"

If $\\alpha$ and $\\beta$ are the roots of the equation $ax^{2}+bx+c=0$, then the equation with roots $\\frac{1}{ \\alpha}$ and $\\frac{1}{ \\beta}$ is

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"NET"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"NET","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/if-alpha-and-beta-are-the-roots-of-the-equation-ax-2-bx-c-0-then-the-equation-with-roots-frac-1-alpha-and-frac-1-beta-is"}],"about":{"@type":"Thing","name":"NET, NUST Entry Test, Mathematics, Quadratic Equations, Quiz"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["NET","NUST Entry Test","Mathematics","Quadratic Equations","Quiz"],"name":"

If $\\alpha$ and $\\beta$ are the roots of the equation $ax^{2}+bx+c=0$, then the equation with roots $\\frac{1}{ \\alpha}$ and $\\frac{1}{ \\beta}$ is

","comment":{"@type":"Comment","text":"

We know that

Sum of roots $= \\alpha + \\beta = \\cfrac{-b}{a} $ and Product of roots $= \\alpha \\beta = \\cfrac{c}{a}$

The roots of the new equation are $\\cfrac{1}{\\alpha}$ and $\\cfrac{1}{\\beta}$

Sum of New Roots $= \\cfrac{1}{\\alpha} + \\cfrac{1}{\\beta}= \\cfrac{\\alpha + \\beta}{\\alpha \\beta} = \\cfrac{\\cfrac{-b}{a}}{\\cfrac{c}{a}}= -\\cfrac{b}{c}$

Product of New Roots $= \\cfrac{1}{\\alpha} \\times \\cfrac{1}{\\beta}= \\cfrac{1}{\\alpha \\beta} = \\cfrac{1}{\\cfrac{c}{a}} = \\cfrac{a}{c}$

Equation with the new roots is given as:

$x^{2} - (\\text{sum of roots})x + (\\text{product of roots}) = 0$

$x^{2} - \\left(-\\cfrac{b}{c}\\right)x + \\cfrac{a}{c} = 0$

$cx^{2} + bx + a = 0$

"},"encodingFormat":"text/markdown","text":"

If $\\alpha$ and $\\beta$ are the roots of the equation $ax^{2}+bx+c=0$, then the equation with roots $\\frac{1}{ \\alpha}$ and $\\frac{1}{ \\beta}$ is

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$\\frac{x^{2}}{ a}+\\frac{x}{ b}+\\frac{1}{ c}=0$

"},{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$ay^{2}+by+c=0$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$ax^{2}+bx+c=0$

"}],"acceptedAnswer":{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$cy^{2}+by+a=0$

","comment":{"@type":"Comment","text":"

We know that

Sum of roots $= \\alpha + \\beta = \\cfrac{-b}{a} $ and Product of roots $= \\alpha \\beta = \\cfrac{c}{a}$

The roots of the new equation are $\\cfrac{1}{\\alpha}$ and $\\cfrac{1}{\\beta}$

Sum of New Roots $= \\cfrac{1}{\\alpha} + \\cfrac{1}{\\beta}= \\cfrac{\\alpha + \\beta}{\\alpha \\beta} = \\cfrac{\\cfrac{-b}{a}}{\\cfrac{c}{a}}= -\\cfrac{b}{c}$

Product of New Roots $= \\cfrac{1}{\\alpha} \\times \\cfrac{1}{\\beta}= \\cfrac{1}{\\alpha \\beta} = \\cfrac{1}{\\cfrac{c}{a}} = \\cfrac{a}{c}$

Equation with the new roots is given as:

$x^{2} - (\\text{sum of roots})x + (\\text{product of roots}) = 0$

$x^{2} - \\left(-\\cfrac{b}{c}\\right)x + \\cfrac{a}{c} = 0$

$cx^{2} + bx + a = 0$

"},"answerExplanation":{"@type":"Comment","text":"

We know that

Sum of roots $= \\alpha + \\beta = \\cfrac{-b}{a} $ and Product of roots $= \\alpha \\beta = \\cfrac{c}{a}$

The roots of the new equation are $\\cfrac{1}{\\alpha}$ and $\\cfrac{1}{\\beta}$

Sum of New Roots $= \\cfrac{1}{\\alpha} + \\cfrac{1}{\\beta}= \\cfrac{\\alpha + \\beta}{\\alpha \\beta} = \\cfrac{\\cfrac{-b}{a}}{\\cfrac{c}{a}}= -\\cfrac{b}{c}$

Product of New Roots $= \\cfrac{1}{\\alpha} \\times \\cfrac{1}{\\beta}= \\cfrac{1}{\\alpha \\beta} = \\cfrac{1}{\\cfrac{c}{a}} = \\cfrac{a}{c}$

Equation with the new roots is given as:

$x^{2} - (\\text{sum of roots})x + (\\text{product of roots}) = 0$

$x^{2} - \\left(-\\cfrac{b}{c}\\right)x + \\cfrac{a}{c} = 0$

$cx^{2} + bx + a = 0$

"}}},"url":"https://classnotes.xyz/mcqs/if-alpha-and-beta-are-the-roots-of-the-equation-ax-2-bx-c-0-then-the-equation-with-roots-frac-1-alpha-and-frac-1-beta-is"}

If $\alpha$ and $\beta$ are the roots of the equation $ax^{2}+bx+c=0$, then the equation with roots $\frac{1}{ \alpha}$ and $\frac{1}{ \beta}$ is