When the parabola touches the X-axis then which of the following condition must be satisfied?

Options

A. `Dgt0`
B. `agt0`
C. D = 0
D. `b^2lt4ac`

Answer

To determine the condition that must be satisfied when a parabola touches the x-axis, we need to analyze the properties of the quadratic equation that represents the parabola. A parabola can be represented by the quadratic equation:

\[ ax^2 + bx + c = 0 \]

The discriminant \( D \) of this quadratic equation is given by:

\[ D = b^2 - 4ac \]

### Step-by-Step Solution:

1. **Understanding the Discriminant**: The discriminant \( D \) helps us determine the nature of the roots of the quadratic equation.
- If \( D > 0 \), the parabola intersects the x-axis at two distinct points (two real roots).
- If \( D = 0 \), the parabola touches the x-axis at exactly one point (one real root, also known as a repeated root).
- If \( D < 0 \), the parabola does not intersect the x-axis at all (no real roots).

2. **Condition for Touching the X-axis**: For the parabola to touch the x-axis, we need it to have exactly one point of contact. This occurs when the discriminant is equal to zero:
\[ D = 0 \implies b^2 - 4ac = 0 \]

3. **Analyzing the Options**:
- **Option 1**: \( D > 0 \) → This means the parabola intersects the x-axis at two points. **Not correct**.
- **Option 2**: \( a > 0 \) → This indicates the parabola opens upwards but does not guarantee it touches the x-axis. **Not necessarily correct**.
- **Option 3**: \( D = 0 \) → This is the condition we derived for the parabola to touch the x-axis. **Correct**.
- **Option 4**: \( b^2 < 4ac \) → This implies \( D < 0 \), meaning the parabola does not touch the x-axis. **Not correct**.

4. **Conclusion**: The correct condition that must be satisfied when a parabola touches the x-axis is:
\[ D = 0 \]

### Final Answer:
The condition that must be satisfied when the parabola touches the x-axis is \( D = 0 \).

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