3. If-1 <m <3 then prove that the roots of x^2 - 2mx + m^2 - 1 = 0 lies in (-2, 4)
Answers
EXPLANATION.
If -1 < m < 3
Prove the roots of equation x² - 2mx + m² - 1 lies in ( -2 , 4).
Range of the equation = ( -2 , 4 ).
Conditions ⇒ a > o satisfy the equation.
(1) = D ≥ 0.
D = b² - 4ac ≥ 0.
⇒ (-2m)² - 4(1)(m² - 1 ) ≥ 0.
⇒ 4m² - 4m² + 4 ≥ 0.
this conditions also satisfy the equation.
⇒ (2) = k₁ < -b/2a < k₂.
⇒ k₁ = -2 and k₂ = 4.
⇒ -2 < -(-2m)/2 < 4.
⇒ -2 < m < 4. .....(1)
This condition also satisfy the equation.
(3) = f(k₁) > 0.
⇒ f(-2) = 0.
put the value of f(-2) in equation we get,
⇒ (-2)² - 2m(-2) + m² - 1 > 0.
⇒ 4 + 4m + m² - 1 > 0.
⇒ m² + 4m + 3 > 0.
⇒ m² + 3m + m + 3 > 0.
⇒ m ( m + 3 ) + 1 ( m + 3 ) > 0.
⇒ ( m + 1 ) ( m + 3 ) > 0.
⇒ m = -1 and -3
put the value it on wavy curve method we get,
m ∈ ( -∞ , -3 ) ∪ ( -1 ,∞ ) .....(2).
(4) = f(k₂) > 0.
⇒ f(4) > 0.
put the value of f(4) in equation we get,
⇒ (4)² - 2m(4) + m² - 1 > 0.
⇒ 16 - 8m + m² - 1 > 0.
⇒ m² - 8m + 15 > 0.
⇒ m² - 5m - 3m + 15 > 0.
⇒ m ( m - 5 ) -3 ( m - 5 ) > 0.
⇒ ( m - 3 ) ( m - 5 ) > 0.
⇒ m = 3 and m = 5.
put the value of m in wavy curve method we get,
m ∈ ( -∞ , 3 ) ∪ ( 5 , ∞ ) .......(3).
From equation (1) , (2) and (3) we can take intersection of the equation.
intersection of equation = (1) ∩ (2) ∩ (3).
we get,
⇒ -1 < m < 3.
HENCE PROVED.
