If the circle `x^2 + y^2 = a^2` cuts off a chord of length `2b` from the line `y = mx +c`, then

Options

A. `sqrt(a^(2)(a+m^(2))) lt c`
B. `sqrt(a^(2)(1-m^(2)))ltc`
C. `sqrt(a^(2)(a+m^(2))) gt c`
D. `sqrt(a^(2)(1-m^(2)))gtc`

Answer

To solve the problem step by step, we will analyze the given equations and derive the necessary relationships.

### Step 1: Write the equations of the circle and the line
The equations given are:
1. Circle: \( x^2 + y^2 = a^2 \)
2. Line: \( y = mx + c \)

### Step 2: Substitute the line equation into the circle equation
Substituting \( y = mx + c \) into the circle equation:
\[ x^2 + (mx + c)^2 = a^2 \]
Expanding the equation:
\[ x^2 + (m^2x^2 + 2mcx + c^2) = a^2 \]
This simplifies to:
\[ (1 + m^2)x^2 + 2mcx + (c^2 - a^2) = 0 \]
This is a quadratic equation in \( x \).

### Step 3: Identify coefficients for the quadratic equation
From the quadratic equation \( Ax^2 + Bx + C = 0 \), we identify:
- \( A = 1 + m^2 \)
- \( B = 2mc \)
- \( C = c^2 - a^2 \)

### Step 4: Use the properties of the roots
Let \( x_1 \) and \( x_2 \) be the roots of the quadratic equation. According to Vieta's formulas:
- Sum of the roots: \( x_1 + x_2 = -\frac{B}{A} = -\frac{2mc}{1 + m^2} \)
- Product of the roots: \( x_1 x_2 = \frac{C}{A} = \frac{c^2 - a^2}{1 + m^2} \)

### Step 5: Calculate the length of the chord
The length of the chord can be calculated using the formula:
\[ \text{Length of chord} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]
Since \( y_1 = mx_1 + c \) and \( y_2 = mx_2 + c \), we have:
\[ y_1 - y_2 = m(x_1 - x_2) \]
Thus, the length of the chord becomes:
\[ \sqrt{(x_1 - x_2)^2 + (m(x_1 - x_2))^2} = \sqrt{(1 + m^2)(x_1 - x_2)^2} \]

### Step 6: Relate the length of the chord to the given length \( 2b \)
Given that the length of the chord is \( 2b \):
\[ \sqrt{(1 + m^2)(x_1 - x_2)^2} = 2b \]
Squaring both sides:
\[ (1 + m^2)(x_1 - x_2)^2 = 4b^2 \]

### Step 7: Calculate \( (x_1 - x_2)^2 \)
From Vieta's formulas, we know:
\[ x_1 - x_2 = \sqrt{(x_1 + x_2)^2 - 4x_1x_2} \]
Substituting the values:
\[ (x_1 - x_2)^2 = \left(-\frac{2mc}{1 + m^2}\right)^2 - 4\left(\frac{c^2 - a^2}{1 + m^2}\right) \]
This leads to:
\[ (x_1 - x_2)^2 = \frac{4m^2c^2}{(1 + m^2)^2} - \frac{4(c^2 - a^2)}{1 + m^2} \]

### Step 8: Substitute into the chord length equation
Substituting this expression back into the chord length equation:
\[ (1 + m^2)\left(\frac{4m^2c^2}{(1 + m^2)^2} - \frac{4(c^2 - a^2)}{1 + m^2}\right) = 4b^2 \]
This simplifies to:
\[ 4m^2c^2 - 4(c^2 - a^2)(1 + m^2) = 4b^2 \]

### Step 9: Final equation
Rearranging gives us the final equation:
\[ m^2c^2 - b^2 - c^2 + a^2 + a^2m^2 = 0 \]