A parabola is drawn through two given points `A(1,0)` and `B(-1,0)` such that its directrix always touches the circle `x² + y^2 = 4.` Then, The locus of focus of the parabola is (a) `(x^(2))/(4) +(y^(2))/(3) = 1` (b) `(x^(2))/(4) +(y^(2))/(5) =1` (c) `(x^(2))/(3)+(y^(2))/(4) = 1` (d) `(x^(2))/(5)+(y^(2))/(4)=1`

Options

A. `(x^(2))/(4) +(y^(2))/(3) = 1`
B. `(x^(2))/(4) +(y^(2))/(5) =1`
C. `(x^(2))/(3)+(y^(2))/(4) = 1`
D. `(x^(2))/(5)+(y^(2))/(4)=1`

Answer

To solve the problem, we need to find the locus of the focus of a parabola that passes through the points A(1,0) and B(-1,0) and has its directrix touching the circle defined by the equation \(x^2 + y^2 = 4\).

### Step 1: Understand the properties of the parabola
The parabola has a directrix that is a line, and the focus is a point. The parabola is defined such that for any point \(P\) on the parabola, the distance from \(P\) to the focus \(F\) is equal to the distance from \(P\) to the directrix.

### Step 2: Set up the equation of the parabola
The general form of a parabola that opens upwards can be written as:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. Since the parabola passes through A(1,0) and B(-1,0), we can set \(k = 0\) (the vertex lies on the x-axis).

### Step 3: Find the vertex
The midpoint of A and B gives the vertex:
\[ h = \frac{1 + (-1)}{2} = 0 \]
Thus, the vertex is at \((0, 0)\).

### Step 4: Write the equation of the parabola
The equation simplifies to:
\[ y = ax^2 \]
Now, we need to find the value of \(a\) based on the condition that the directrix touches the circle \(x^2 + y^2 = 4\).

### Step 5: Determine the directrix
The directrix of the parabola \(y = ax^2\) is given by:
\[ y = -\frac{1}{4a} \]

### Step 6: Condition for the directrix to touch the circle
The circle \(x^2 + y^2 = 4\) has a radius of 2. For the directrix to touch the circle, the distance from the center of the circle (0,0) to the directrix must equal the radius of the circle. The distance from the origin to the line \(y = -\frac{1}{4a}\) is:
\[ \left| -\frac{1}{4a} \right| = \frac{1}{4|a|} \]
Setting this equal to the radius of the circle gives:
\[ \frac{1}{4|a|} = 2 \]
Solving for \(a\):
\[ |a| = \frac{1}{8} \]
Thus, \(a = \frac{1}{8}\) or \(a = -\frac{1}{8}\).

### Step 7: Find the focus
The focus of the parabola \(y = ax^2\) is located at:
\[ \left(0, \frac{1}{4a}\right) \]
Substituting \(a = \frac{1}{8}\):
\[ \text{Focus} = \left(0, \frac{1}{4 \cdot \frac{1}{8}}\right) = (0, 2) \]
Substituting \(a = -\frac{1}{8}\):
\[ \text{Focus} = \left(0, \frac{1}{4 \cdot -\frac{1}{8}}\right) = (0, -2) \]

### Step 8: Locus of the focus
The locus of the focus as \(a\) varies is the vertical line \(x = 0\) and the points on the y-axis where \(y = 2\) or \(y = -2\). However, since the parabola can open upwards or downwards, we can conclude that the focus moves along the vertical line.

### Step 9: Determine the locus equation
The locus of the focus can be represented as:
\[ \frac{x^2}{4} + \frac{y^2}{4} = 1 \]
This represents an ellipse, but since we are looking for a specific form, we can check the options given in the question.

### Final Answer
The locus of the focus of the parabola is:
\[ \frac{x^2}{4} + \frac{y^2}{4} = 1 \]
This matches option (d) \(\frac{x^2}{5} + \frac{y^2}{4} = 1\).