A parabola has its latus rectum along PQ, where `P(x_1, y_1)` and `Q(x_2, y_2), y_1 gt0, y_2 gt0` are the end points of the latus rectums of the ellipse `x^2/4+y^2/1=1` . Coordinates of the focus of the parabola are

Options

A. (0,-1/2)
B. (0,0)
C. (0,1/2)
D. (0,1)

Answer

To solve the problem, we need to find the coordinates of the focus of a parabola whose latus rectum is along the endpoints \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) of the latus rectum of the given ellipse \( \frac{x^2}{4} + \frac{y^2}{1} = 1 \).

### Step 1: Identify the parameters of the ellipse
The equation of the ellipse is given as:
\[ \frac{x^2}{4} + \frac{y^2}{1} = 1 \]
From this, we can identify:
- \( a^2 = 4 \) (major semi-axis)
- \( b^2 = 1 \) (minor semi-axis)

Thus, \( a = 2 \) and \( b = 1 \).

### Step 2: Calculate the foci of the ellipse
The foci of the ellipse can be calculated using the formula:
\[ c = \sqrt{a^2 - b^2} = \sqrt{4 - 1} = \sqrt{3} \]
The coordinates of the foci are:
\[ F_1(c, 0) = (\sqrt{3}, 0) \quad \text{and} \quad F_2(-c, 0) = (-\sqrt{3}, 0) \]

### Step 3: Determine the endpoints of the latus rectum
The endpoints of the latus rectum of the ellipse can be found using the formula:
\[ \left( \pm \frac{b^2}{a}, \pm a \right) \]
Substituting the values of \( a \) and \( b \):
\[ \left( \pm \frac{1}{2}, \pm 2 \right) \]
Since \( y_1 > 0 \) and \( y_2 > 0 \), we take the positive coordinates:
\[ P\left(\frac{1}{2}, 2\right) \quad \text{and} \quad Q\left(-\frac{1}{2}, 2\right) \]

### Step 4: Find the coordinates of the focus of the parabola
The focus of the parabola is the midpoint of the line segment joining points \( P \) and \( Q \):
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of \( P \) and \( Q \):
\[ \text{Midpoint} = \left( \frac{\frac{1}{2} + \left(-\frac{1}{2}\right)}{2}, \frac{2 + 2}{2} \right) = \left( 0, 2 \right) \]

### Conclusion
The coordinates of the focus of the parabola are:
\[ \boxed{(0, 2)} \]