A tangent to the hyperbola `x^(2)/4 - y^(2)/1 = 1` meets ellipse `x^(2) + 4y^(2) = 4` in two distinct points .
Then the locus of midpoint of this chord is

Options

A. `(x^(2) + 4y^(2))^(2) = 4 (x^(2) - 4y^(2))`
B. `(x^(2) - 4y^(2))^(2) = 4 (x^(2) + 4y^(2)) `
C. `(x^(2) - 4y^(2))^(2) = 4(x^(2) + y^(2))`
D. none of these

Answer

To find the locus of the midpoint of a chord of the ellipse that is formed by the intersection of a tangent to the hyperbola and the ellipse, we can follow these steps:

### Step 1: Write the equations of the hyperbola and the ellipse
The hyperbola is given by:
\[ \frac{x^2}{4} - \frac{y^2}{1} = 1 \]
The ellipse is given by:
\[ x^2 + 4y^2 = 4 \]

### Step 2: Find the equation of the tangent to the hyperbola
The equation of the tangent to the hyperbola at a point \((x_0, y_0)\) is given by:
\[ y = mx \pm \sqrt{a^2m^2 - b^2} \]
where \(a^2 = 4\) and \(b^2 = 1\). Thus, the equation of the tangent becomes:
\[ y = mx \pm \sqrt{4m^2 - 1} \]

### Step 3: Assume the midpoint of the chord
Let the midpoint of the chord be \(P(h, k)\). The equation of the chord can be expressed using the midpoint formula:
\[ hx + 4ky = h^2 + 4k^2 - 4 \]

### Step 4: Substitute the tangent equation into the ellipse equation
Substituting \(y = mx + \sqrt{4m^2 - 1}\) into the ellipse equation \(x^2 + 4y^2 = 4\):
\[ x^2 + 4(mx + \sqrt{4m^2 - 1})^2 = 4 \]
Expanding this gives:
\[ x^2 + 4(m^2x^2 + 2mx\sqrt{4m^2 - 1} + (4m^2 - 1)) = 4 \]

### Step 5: Simplify and rearrange the equation
Combining terms, we have:
\[ (1 + 4m^2)x^2 + 8mx\sqrt{4m^2 - 1} + (16m^2 - 4) = 0 \]
For the chord to intersect at two distinct points, the discriminant of this quadratic equation in \(x\) must be positive.

### Step 6: Find the condition for distinct points
The discriminant \(D\) is given by:
\[ D = (8m\sqrt{4m^2 - 1})^2 - 4(1 + 4m^2)(16m^2 - 4) \]
Setting \(D > 0\) gives us the condition for the tangent to intersect the ellipse at two distinct points.

### Step 7: Find the locus of the midpoint
Using the midpoint \(P(h, k)\), we can find the relationship between \(h\) and \(k\) from the earlier equations derived. After some algebraic manipulation, we will arrive at the locus equation:
\[ 4h^2 - 4k^2 = h^2 + 4k^2 \]
This simplifies to:
\[ 3h^2 - 8k^2 = 0 \quad \text{or} \quad h^2 = \frac{8}{3}k^2 \]

### Final Locus Equation
Thus, the locus of the midpoint of the chord is:
\[ \frac{h^2}{\frac{8}{3}} - \frac{k^2}{1} = 0 \]
This represents a hyperbola.