The ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` and the straight line `y=mx+c` intersect in real points only if:

Options

A. `a^(2)m^(2) lt c^(2) - b^(2)`
B. `a^(2)m^(2) gt c^(2) - b^(2)`
C. `a^(2)m^(2) ge c^(2) - b^(2)`
D. `c ge b`

Answer

To solve the problem of determining the condition under which the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and the straight line \(y = mx + c\) intersect in real points, we can follow these steps:

### Step 1: Substitute the line equation into the ellipse equation
We start by substituting \(y = mx + c\) into the ellipse equation:

\[ \frac{x^2}{a^2} + \frac{(mx + c)^2}{b^2} = 1 \]

### Step 2: Expand the equation
Expanding the equation gives:

\[ \frac{x^2}{a^2} + \frac{m^2x^2 + 2mcx + c^2}{b^2} = 1 \]

### Step 3: Combine terms
To combine the terms, we can multiply through by \(a^2b^2\) to eliminate the denominators:

\[ b^2x^2 + a^2(m^2x^2 + 2mcx + c^2) = a^2b^2 \]

This simplifies to:

\[ (b^2 + a^2m^2)x^2 + 2a^2mcx + (a^2c^2 - a^2b^2) = 0 \]

### Step 4: Identify coefficients of the quadratic equation
This is a quadratic equation in the form \(Ax^2 + Bx + C = 0\), where:
- \(A = b^2 + a^2m^2\)
- \(B = 2a^2mc\)
- \(C = a^2c^2 - a^2b^2\)

### Step 5: Use the discriminant condition
For the quadratic equation to have real solutions, the discriminant must be non-negative:

\[ D = B^2 - 4AC \geq 0 \]

Substituting the values of \(A\), \(B\), and \(C\):

\[ (2a^2mc)^2 - 4(b^2 + a^2m^2)(a^2c^2 - a^2b^2) \geq 0 \]

### Step 6: Simplify the discriminant
Calculating the discriminant:

\[ 4a^4m^2c^2 - 4(b^2 + a^2m^2)(a^2c^2 - a^2b^2) \geq 0 \]

Dividing through by 4:

\[ a^4m^2c^2 - (b^2 + a^2m^2)(a^2c^2 - a^2b^2) \geq 0 \]

### Step 7: Expand and rearrange
Expanding the second term gives:

\[ a^4m^2c^2 - (a^2b^2m^2 + b^2c^2 - b^4) \geq 0 \]

Rearranging leads to:

\[ a^4m^2c^2 + b^2b^4 - b^2c^2 - a^2b^2m^2 \geq 0 \]

### Step 8: Factor out common terms
Factoring out \(b^2\):

\[ b^2(m^2a^2 + b^2) - c^2 \geq 0 \]

### Step 9: Final condition
This gives us the final condition for the intersection of the ellipse and the line:

\[ a^2m^2 \geq c^2 - b^2 \]

### Conclusion
Thus, the ellipse and the line intersect in real points only if:

\[ a^2m^2 \geq c^2 - b^2 \]