if the line `4x +3y +1=0` meets the parabola `y^2=8x` then the mid point of the chord is

Options

A. `(2,- 3)`
B. `(2,4)`
C. `(5//2, 14//3)`
D. `(5, 8)`

Explanation

To find the midpoint of the chord formed by the intersection of the line \(4x + 3y + 1 = 0\) and the parabola \(y^2 = 8x\), we can follow these steps:

### Step 1: Express \(y\) in terms of \(x\) from the line equation
The line equation is given as:
\[ 4x + 3y + 1 = 0 \]
We can rearrange this to express \(y\):
\[ 3y = -4x - 1 \implies y = -\frac{4}{3}x - \frac{1}{3} \]

**Hint:** Rearranging the equation helps isolate one variable, making substitution easier.

### Step 2: Substitute \(y\) into the parabola equation
Now, substitute \(y\) from the line equation into the parabola equation \(y^2 = 8x\):
\[ \left(-\frac{4}{3}x - \frac{1}{3}\right)^2 = 8x \]
Expanding the left side:
\[ \left(\frac{16}{9}x^2 + \frac{8}{9}x + \frac{1}{9}\right) = 8x \]

**Hint:** Squaring a binomial involves applying the formula \((a + b)^2 = a^2 + 2ab + b^2\).

### Step 3: Clear the fraction and rearrange the equation
Multiply through by 9 to eliminate the denominator:
\[ 16x^2 + 8x + 1 = 72x \]
Rearranging gives:
\[ 16x^2 - 64x + 1 = 0 \]

**Hint:** Clearing fractions simplifies the equation and makes it easier to work with.

### Step 4: Use the quadratic formula to find \(x\)
We can find the roots \(x_1\) and \(x_2\) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 16\), \(b = -64\), and \(c = 1\):
\[ x = \frac{64 \pm \sqrt{(-64)^2 - 4 \cdot 16 \cdot 1}}{2 \cdot 16} \]
Calculating the discriminant:
\[ x = \frac{64 \pm \sqrt{4096 - 64}}{32} = \frac{64 \pm \sqrt{4032}}{32} \]

**Hint:** The quadratic formula is a powerful tool for finding roots of any quadratic equation.

### Step 5: Find the sum of the roots
From Vieta's formulas, the sum of the roots \(x_1 + x_2\) is given by:
\[ x_1 + x_2 = -\frac{b}{a} = \frac{64}{16} = 4 \]

**Hint:** Vieta's formulas relate the coefficients of the polynomial to sums and products of its roots.

### Step 6: Find the midpoint of the chord
The x-coordinate of the midpoint \(M\) is:
\[ M_x = \frac{x_1 + x_2}{2} = \frac{4}{2} = 2 \]

**Hint:** The midpoint formula for coordinates is simply the average of the x-coordinates and y-coordinates of the endpoints.

### Step 7: Find the y-coordinate of the midpoint
To find the y-coordinate of the midpoint, we can use the line equation to find \(y\) when \(x = 2\):
\[ y = -\frac{4}{3}(2) - \frac{1}{3} = -\frac{8}{3} - \frac{1}{3} = -\frac{9}{3} = -3 \]

**Hint:** Substituting the x-coordinate back into the line equation gives the corresponding y-coordinate.

### Final Answer
Thus, the midpoint of the chord is:
\[ \boxed{(2, -3)} \]