The locus of mid-point of family of chords `lamda x + y - 5 =0` (parameter) of the parabola `x ^(2) = 20y` is

Options

A. `y ^(2) = 10(x-5)`
B. `x ^(2) = 10 (y -5)`
C. `x ^(2) +y ^(2) = 25`
D. none of these

Answer

To find the locus of the mid-point of the family of chords given by the equation \( \lambda x + y - 5 = 0 \) of the parabola \( x^2 = 20y \), we can follow these steps:

### Step 1: Rewrite the chord equation
The equation of the chord can be rearranged as:
\[ y = 5 - \lambda x \]

### Step 2: Identify the points of intersection
To find the points where this chord intersects the parabola \( x^2 = 20y \), we substitute \( y \) from the chord equation into the parabola equation:
\[ x^2 = 20(5 - \lambda x) \]
This simplifies to:
\[ x^2 = 100 - 20\lambda x \]
Rearranging gives us:
\[ x^2 + 20\lambda x - 100 = 0 \]

### Step 3: Find the mid-point of the chord
The roots of the quadratic equation represent the x-coordinates of the points where the chord intersects the parabola. The mid-point \( M \) of the chord can be found using the formula for the mid-point of the roots:
\[ M_x = -\frac{b}{2a} = -\frac{20\lambda}{2} = -10\lambda \]
Now, substituting \( M_x \) back into the chord equation to find \( M_y \):
\[ M_y = 5 - \lambda(-10\lambda) = 5 + 10\lambda^2 \]

### Step 4: Express \( M \) in terms of \( \lambda \)
Thus, the coordinates of the mid-point \( M \) are:
\[ M(-10\lambda, 5 + 10\lambda^2) \]

### Step 5: Eliminate the parameter \( \lambda \)
To find the locus, we need to eliminate \( \lambda \). From \( M_x = -10\lambda \), we can express \( \lambda \) as:
\[ \lambda = -\frac{M_x}{10} \]
Substituting this into the equation for \( M_y \):
\[ M_y = 5 + 10\left(-\frac{M_x}{10}\right)^2 \]
This simplifies to:
\[ M_y = 5 + 10\left(\frac{M_x^2}{100}\right) = 5 + \frac{M_x^2}{10} \]

### Step 6: Rearranging to find the locus equation
Rearranging gives:
\[ M_y - 5 = \frac{M_x^2}{10} \]
Multiplying through by 10 yields:
\[ 10(M_y - 5) = M_x^2 \]
Thus, we have:
\[ M_x^2 = 10M_y - 50 \]
This is the equation of the locus of the mid-point of the chords.

### Final Result
The locus of the mid-point of the family of chords is:
\[ x^2 = 10y - 50 \]