If any tangent to the parabola `x^(2) = 4y` intersects the hyperbola xy = 2 at two points P and Q, then the mid point of line segment PQ lies on a parabola with axis along:

Options

A. x-axis and focus on positive x-axis
B. y-axis and focus on positive y-axis
C. x-axis and focus on negative x-axis
D. y-axis and focus on negative y-axis

Answer

To solve the problem step by step, we need to find the locus of the midpoints of the line segments formed by the intersection of the tangent to the parabola \(x^2 = 4y\) and the hyperbola \(xy = 2\).

### Step 1: Equation of the Tangent to the Parabola
The equation of the parabola is given by \(x^2 = 4y\). The general form of the tangent to this parabola at a point \((x_1, y_1)\) is given by:
\[ y = mx - \frac{m^2}{4} \]
where \(m\) is the slope of the tangent.

### Step 2: Substitute the Tangent Equation into the Hyperbola
We need to find the points of intersection of the tangent line with the hyperbola \(xy = 2\). Substituting \(y = mx - \frac{m^2}{4}\) into the hyperbola equation:
\[ x(mx - \frac{m^2}{4}) = 2 \]
This simplifies to:
\[ mx^2 - \frac{m^2}{4}x - 2 = 0 \]

### Step 3: Find the Roots of the Quadratic Equation
The roots of the quadratic equation \(mx^2 - \frac{m^2}{4}x - 2 = 0\) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = m\), \(b = -\frac{m^2}{4}\), and \(c = -2\). Thus,
\[ x = \frac{\frac{m^2}{4} \pm \sqrt{\left(-\frac{m^2}{4}\right)^2 - 4 \cdot m \cdot (-2)}}{2m} \]
Calculating the discriminant:
\[ \left(-\frac{m^2}{4}\right)^2 - 4 \cdot m \cdot (-2) = \frac{m^4}{16} + 8m \]

### Step 4: Calculate the Midpoint of Points P and Q
Let the roots of the quadratic be \(x_1\) and \(x_2\). The midpoint \(M\) of the line segment \(PQ\) is given by:
\[ M_x = \frac{x_1 + x_2}{2} = \frac{\frac{m^2}{4m}}{2} = \frac{m}{8} \]
For the y-coordinates, we substitute \(x_1\) and \(x_2\) back into the tangent equation:
\[ M_y = \frac{y_1 + y_2}{2} = \frac{m\left(\frac{m}{8}\right) - \frac{m^2}{4}}{2} = -\frac{m^2}{16} \]

### Step 5: Find the Locus of the Midpoint
Now we have the coordinates of the midpoint:
\[ M\left(\frac{m}{8}, -\frac{m^2}{16}\right) \]
Let \(h = \frac{m}{8}\) and \(k = -\frac{m^2}{16}\). We can express \(m\) in terms of \(h\):
\[ m = 8h \]
Substituting this into the equation for \(k\):
\[ k = -\frac{(8h)^2}{16} = -\frac{64h^2}{16} = -4h^2 \]
This gives us the equation of the parabola:
\[ 4h^2 + k = 0 \quad \text{or} \quad k = -4h^2 \]

### Conclusion
The locus of the midpoints \(M\) lies on the parabola \(k = -4h^2\), which has its axis along the y-axis.