Theory of EquationsexpertFree

Theory of Equations: Roots Equation Roots Equation

JEE Maths question with a full step-by-step solution.

Question
If roots of the equation a(x−ℓ)(x−m)+n=0a(x-\ell)(x-m) + n = 0 are α,β\alpha, \beta, then roots of the equation a ⁣(xx−k−α) ⁣(xx−k−β)−n=0a\!\left(\dfrac{x}{x-k} - \alpha\right)\!\left(\dfrac{x}{x-k} - \beta\right) - n = 0 are
Aℓℓ+1, mm+1\dfrac{\ell}{\ell+1},\, \dfrac{m}{m+1}
Bk(ℓ+1)ℓ, k(m+1)m\dfrac{k(\ell+1)}{\ell},\, \dfrac{k(m+1)}{m}
Ckℓℓ−1, kmm−1\dfrac{k\ell}{\ell-1},\, \dfrac{km}{m-1}correct
Dℓ−1ℓ, m−1m\dfrac{\ell-1}{\ell},\, \dfrac{m-1}{m}
Solution
Step 1: Since α,β\alpha, \beta are roots of a(t−ℓ)(t−m)+n=0a(t-\ell)(t-m)+n = 0, we have a(t−ℓ)(t−m)+n=a(t−α)(t−β)a(t-\ell)(t-m)+n = a(t-\alpha)(t-\beta). Therefore a(t−α)(t−β)−n=a(t−ℓ)(t−m)a(t-\alpha)(t-\beta) - n = a(t-\ell)(t-m). Step 2: Set t=xx−kt = \dfrac{x}{x-k}. The equation becomes:
a ⁣(xx−k−α) ⁣(xx−k−β)−n=0  ⟹  a ⁣(xx−k−ℓ) ⁣(xx−k−m)=0a\!\left(\frac{x}{x-k}-\alpha\right)\!\left(\frac{x}{x-k}-\beta\right) - n = 0 \implies a\!\left(\frac{x}{x-k}-\ell\right)\!\left(\frac{x}{x-k}-m\right) = 0
 Step 3: xx−k=ℓ  ⟹  x=ℓ(x−k)  ⟹  x=kℓℓ−1\dfrac{x}{x-k} = \ell \implies x = \ell(x-k) \implies x = \dfrac{k\ell}{\ell-1}. Similarly: xx−k=m  ⟹  x=kmm−1\dfrac{x}{x-k} = m \implies x = \dfrac{km}{m-1}. Correct answer: (3)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Theory of Equations · easy
The values of α\alpha and β\beta for which the quadratic equation x2+2x+2+e2α−cos⁡β=0x^2 + 2x + 2 + e^{2\alpha} - \cos\beta = 0 has a real solution is
Theory of Equations · easy
If the vertex of the quadratic expression y=2x2−4x+6y = 2x^2 - 4x + 6 is (m,n)(m, n), then the quadratic equation whose vertex is (2m,n2)\left(2m, \dfrac{n}{2}\right) is
Theory of Equations · easy
If α,β\alpha, \beta are the roots of x2−p(x+1)−c=0x^2 - p(x+1) - c = 0, c≠1c \neq 1, then the value of (α+1)(β+1)(\alpha+1)(\beta+1) is equal to
Theory of Equations · easy
Let α\alpha and β\beta be the roots of the equation x2−(p−2)x−(p−1)=0x^2 - (p-2)x - (p-1) = 0, p∈Rp \in \mathbb{R}. If α2+β2\alpha^2 + \beta^2 is least, then pp equals

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath