IF (a,b) is the mid point of chord passing through the vertex of the parabola `y^2=4x` , then

Options

A. a=2b
B. 2a=b
C. `a^2=2b`
D. `2a=b^2`

Answer

To solve the problem step by step, we will derive the relationship between \(a\) and \(b\) given that \((a, b)\) is the midpoint of a chord passing through the vertex of the parabola \(y^2 = 4x\).

### Step-by-Step Solution:

1. **Identify the Vertex of the Parabola**:
The given parabola is \(y^2 = 4x\). The vertex of this parabola is at the origin, which is the point \((0, 0)\).

2. **Parameterize a Point on the Parabola**:
A general point \(P\) on the parabola can be expressed in terms of a parameter \(t\):
\[ P(t) = (t^2, 2t) \]
Here, \(t^2\) is the x-coordinate and \(2t\) is the y-coordinate of the point on the parabola.

3. **Find the Midpoint of the Chord**:
The chord passes through the vertex \((0, 0)\) and the point \(P(t)\). The midpoint \(M\) of the chord connecting the vertex and the point \(P(t)\) is calculated as follows:
\[ M = \left(\frac{0 + t^2}{2}, \frac{0 + 2t}{2}\right) = \left(\frac{t^2}{2}, t\right) \]

4. **Relate the Midpoint to Given Coordinates**:
We are given that the midpoint \(M\) is \((a, b)\). Therefore, we can equate the coordinates:
\[ a = \frac{t^2}{2} \quad \text{and} \quad b = t \]

5. **Express \(t\) in terms of \(b\)**:
From the second equation, we have:
\[ t = b \]

6. **Substitute \(t\) into the equation for \(a\)**:
Substitute \(t = b\) into the equation for \(a\):
\[ a = \frac{b^2}{2} \]

7. **Rearrange to find the relationship**:
Rearranging the equation gives:
\[ b^2 = 2a \]

### Final Relationship:
The relationship between \(a\) and \(b\) is:
\[ b^2 = 2a \]