If the tangent at the point P `(2,4)` to the parabola `y^2 = 8x` meets the parabola `y^2 = 8x+5` at Q and R then the mid-point of QR is

Options

A. `(4, 2)`
B. `(2, 4)`
C. `(7,9)`
D. none

Answer

To solve the problem step-by-step, we will find the tangent at point P(2, 4) to the parabola \( y^2 = 8x \), determine where this tangent intersects the second parabola \( y^2 = 8x + 5 \), and then find the midpoint of the intersection points Q and R.

### Step 1: Find the equation of the tangent at point P(2, 4)

For the parabola \( y^2 = 8x \), the general formula for the tangent at a point \( (x_1, y_1) \) is given by:

\[ yy_1 = 4(x + x_1) \]

Substituting \( (x_1, y_1) = (2, 4) \):

\[ y \cdot 4 = 4(x + 2) \]

This simplifies to:

\[ 4y = 4x + 8 \quad \Rightarrow \quad y = x + 2 \]

### Step 2: Substitute the tangent equation into the second parabola

Now we need to find where this tangent line \( y = x + 2 \) intersects the second parabola \( y^2 = 8x + 5 \). We substitute \( y \) from the tangent equation into the second parabola's equation:

\[ (x + 2)^2 = 8x + 5 \]

Expanding the left side:

\[ x^2 + 4x + 4 = 8x + 5 \]

### Step 3: Rearranging the equation

Rearranging gives us:

\[ x^2 + 4x + 4 - 8x - 5 = 0 \]

This simplifies to:

\[ x^2 - 4x - 1 = 0 \]

### Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1, b = -4, c = -1 \):

\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]

Calculating the discriminant:

\[ x = \frac{4 \pm \sqrt{16 + 4}}{2} = \frac{4 \pm \sqrt{20}}{2} = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5} \]

### Step 5: Find the corresponding y-coordinates

Now we find the y-coordinates corresponding to these x-values using the tangent equation \( y = x + 2 \):

1. For \( x = 2 + \sqrt{5} \):

\[ y = (2 + \sqrt{5}) + 2 = 4 + \sqrt{5} \]

2. For \( x = 2 - \sqrt{5} \):

\[ y = (2 - \sqrt{5}) + 2 = 4 - \sqrt{5} \]

Thus, the points of intersection Q and R are:

- \( Q(2 + \sqrt{5}, 4 + \sqrt{5}) \)
- \( R(2 - \sqrt{5}, 4 - \sqrt{5}) \)

### Step 6: Find the midpoint of points Q and R

The midpoint \( M \) of points Q and R is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting the coordinates of Q and R:

\[ M = \left( \frac{(2 + \sqrt{5}) + (2 - \sqrt{5})}{2}, \frac{(4 + \sqrt{5}) + (4 - \sqrt{5})}{2} \right) \]

Calculating the x-coordinate:

\[ M_x = \frac{4}{2} = 2 \]

Calculating the y-coordinate:

\[ M_y = \frac{8}{2} = 4 \]

Thus, the midpoint of QR is:

\[ M(2, 4) \]

### Final Answer

The midpoint of QR is \( (2, 4) \).

---