SATHEE: Theory of Equations 5 Question 5

Theory of Equations 5 Question 5

5. Let $a, b, c$ be real numbers, $a \neq 0$ . If $α$ is a root of $a^{2} x^{2} + b x + c = 0, β$ is the root of $a^{2} x^{2} - b x - c = 0$ and $0 < α < β$ , then the equation $a^{2} x^{2} + 2 b x + 2 c = 0$ has a root $γ$ that always satisfies

$(1989, 2 M)$

(a) $γ = \frac{α + β}{2}$

(b) $γ = α + \frac{β}{2}$

(d) $α < γ < β$

Show Answer

Answer:

Correct Answer: 5. (d)

Solution:

Since, $α$ is a root of $a^{2} x^{2} + b x + c = 0$

$\Rightarrow a^{2} α^{2} + b α + c = 0$

and $β$ is a root of $a^{2} x^{2} - b x - c = 0$

$\Rightarrow a^{2} β^{2} - b β - c = 0$

Let $f (x) = a^{2} x^{2} + 2 b x + 2 c$

$∴ f (α) = a^{2} α^{2} + 2 b α + 2 c$

$= a^{2} α^{2} - 2 a^{2} α^{2} = - a^{2} α^{2}$

[from Eq. (i)]

and $f (β) = α^{2} β^{2} + 2 b β + 2 c$

$= a^{2} β^{2} + 2 a^{2} β^{2} = 3 a^{2} β^{2} [from Eq. (ii)]$

$\Rightarrow f (α) f (β) < 0$

$f (x)$ must have a root lying in the open interval $(α, β)$ .

$∴ α < γ < β$

Theory of Equations 5 Question 5

5. Let $a, b, c$ be real numbers, $a \neq 0$ . If $α$ is a root of $a^{2} x^{2} + b x + c = 0, β$ is the root of $a^{2} x^{2} - b x - c = 0$ and $0 < α < β$ , then the equation $a^{2} x^{2} + 2 b x + 2 c = 0$ has a root $γ$ that always satisfies

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Theory of Equations 5 Question 5

5. Let a,b,c be real numbers, a≠0. If α is a root of a2x2+bx+c=0,β is the root of a2x2−bx−c=0 and 0<α<β, then the equation a2x2+2bx+2c=0 has a root γ that always satisfies

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

5. Let $a, b, c$ be real numbers, $a \neq 0$ . If $α$ is a root of $a^{2} x^{2} + b x + c = 0, β$ is the root of $a^{2} x^{2} - b x - c = 0$ and $0 < α < β$ , then the equation $a^{2} x^{2} + 2 b x + 2 c = 0$ has a root $γ$ that always satisfies