The Values of a for Which Both the Roots of the Equation 1-a^2x^2-2ax-10 Lie Between 0 and q
In this article, we will explore how to determine the range of values for the parameter a such that both roots of the quadratic equation 1 - a^2x^2 - 2ax - 1 0 lie within the interval (0, q). This problem involves the analysis of the location of roots and interval constraints on the roots.
1 - a^2x^2 - 2ax - 1 0
The given quadratic equation is:
1 - a^2x^2 - 2ax - 1 0
First, let's simplify and rewrite the equation for clarity:
1 - a^2x^2 - 2ax - 1 0
> -a^2x^2 - 2ax 0 0
> a^2x^2 2ax - 1 0
To solve this equation, we use the quadratic formula, which states that for an equation of the form ax^2 bx c 0, the roots are given by:
x u00BD(-b u00B1 u221A(b^2 - 4ac))
Solving the Quadratic Equation
For our equation a^2x^2 2ax - 1 0, we identify the coefficients:
a a^2, b 2a, c -1
Plugging these values into the quadratic formula, we get:
x u00BD(-2a u00B1 u221A((2a)^2 - 4(a^2)(-1)))
> x u00BD(-2a u00B1 u221A(4a^2 4a^2))
> x u00BD(-2a u00B1 u221A(8a^2))
> x u00BD(-2a u00B1 2u221A2a)
Thus, the roots are:
x1 u00BD(-2a 2u221A2a) 1/u00BD(1 - u221A2a)
x2 u00BD(-2a - 2u221A2a) 1/u00BD(1 u221A2a)
For both roots to lie between 0 and q, the following conditions must be satisfied:
Interval Constraints on the Roots
For x1 and x2 to lie in the interval (0, q), we need:
x1 > 0 x1 x2 > 0 x2Let's analyze each condition:
Condition 1: x1 > 0
For x1 1/u00BD(1 - u221A2a) > 0, the denominator (1 - u221A2a) must be positive:
1 - u221A2a > 0
u221A2a
u221A2a
a
a
Condition 2: x1
For x1 1/u00BD(1 - u221A2a) :
1/u00BD(1 - u221A2a)
u00BD(1 - u221A2a)
1 - u221A2a
u221A2a > 1 - 2/q
2a > (1 - 2/q)^2
a > u00BD((1 - 2/q)^2)
a > u00BD(1 - 4/q 4/q^2)
a > u00BD(1 - 4/q 4/q^2)
Condition 3: x2 > 0
For x2 1/u00BD(1 u221A2a) > 0, the denominator (1 u221A2a) must be positive:
1 u221A2a > 0
u221A2a > -1
a > -1/u221A2
a > -u221A2/2
Condition 4: x2
For x2 4(u00BD(1 u221A2a)) :
u00BD(1 u221A2a)
1 u221A2a
u221A2a
2a
a
a
Combining the conditions, we find that the value of a must satisfy:
WARN/r/n 1/u221A2
WARN/r/n [1/u221A2, q]
WARN/r/n a > u00BD((1 - 2/q)^2)
WARN/r/n a
Thus, the range of values for a such that both roots of the quadratic equation lie within the interval (0, q) is given by:
u00BD((1 - 2/q)^2)
Conclusion
In conclusion, the values of a for which both roots of the quadratic equation 1 - a^2x^2 - 2ax - 1 0 lie between 0 and q can be determined by analyzing the conditions on the roots. These conditions ensure that both roots are positive and lie within the specified interval. By applying the interval constraints and solving the inequalities, we obtain the final range for a.
References
For further reading on the topic, refer to:
Algebra and Trigonometry by Ron Larson. (Chapter on Quadratic Equations) Calculus by James Stewart. (Section on Roots of Polynomials)