The quadratic equation ` (1 + a^(2)) x^(2) + 2 abx + ( b^(2) - c^(2)) = 0 ` has only one root . What is the value of ` c^(2) (1 + a^(2))` ?

Options

A. `a^(2) `
B. ` b^(2)`
C. ` c^(2)`
D. ab

Answer

To solve the problem, we need to analyze the given quadratic equation:

\[ (1 + a^2)x^2 + 2abx + (b^2 - c^2) = 0 \]

We know that a quadratic equation has only one root when its discriminant is equal to zero. The discriminant \(D\) for a quadratic equation of the form \(Ax^2 + Bx + C = 0\) is given by:

\[ D = B^2 - 4AC \]

In our case:
- \(A = 1 + a^2\)
- \(B = 2ab\)
- \(C = b^2 - c^2\)

Now, we will calculate the discriminant:

\[ D = (2ab)^2 - 4(1 + a^2)(b^2 - c^2) \]

Calculating \(D\):

1. First, calculate \(B^2\):
\[ (2ab)^2 = 4a^2b^2 \]

2. Next, calculate \(4AC\):
\[ 4(1 + a^2)(b^2 - c^2) = 4(1 + a^2)b^2 - 4(1 + a^2)c^2 \]

So, we can rewrite the discriminant as:

\[ D = 4a^2b^2 - (4(1 + a^2)b^2 - 4(1 + a^2)c^2) \]

3. Simplifying \(D\):
\[ D = 4a^2b^2 - 4(1 + a^2)b^2 + 4(1 + a^2)c^2 \]
\[ D = 4a^2b^2 - 4b^2 - 4a^2b^2 + 4(1 + a^2)c^2 \]
\[ D = -4b^2 + 4(1 + a^2)c^2 \]

4. Setting the discriminant to zero for the equation to have only one root:
\[ -4b^2 + 4(1 + a^2)c^2 = 0 \]

5. Dividing the entire equation by 4:
\[ -b^2 + (1 + a^2)c^2 = 0 \]

6. Rearranging gives us:
\[ (1 + a^2)c^2 = b^2 \]

Thus, the value of \(c^2(1 + a^2)\) is equal to \(b^2\).

**Final Answer:**
\[ c^2(1 + a^2) = b^2 \]