The line `y = mx` intersects the circle `x^(2)+y^(2) -2x - 2y = 0` and `x^(2)+y^(2) +6x - 8y =0` at point A and B (points being other than origin). The range of m such that origin divides AB internally is

Options

A. `-1 lt m lt (3)/(4)`
B. `m gt (4)/(3)` or `m lt -2`
C. `-2 lt m lt (4)/(3)`
D. `m gt -1`

Explanation

To solve the problem, we need to find the range of the slope \( m \) such that the line \( y = mx \) intersects the two given circles and the origin divides the line segment \( AB \) internally.

### Step 1: Rewrite the equations of the circles
The equations of the circles are:
1. \( x^2 + y^2 - 2x - 2y = 0 \)
2. \( x^2 + y^2 + 6x - 8y = 0 \)

We can rewrite these equations in standard form.

For the first circle:
\[ x^2 - 2x + y^2 - 2y = 0 \implies (x-1)^2 + (y-1)^2 = 2 \]
This represents a circle centered at \( (1, 1) \) with a radius of \( \sqrt{2} \).

For the second circle:
\[ x^2 + 6x + y^2 - 8y = 0 \implies (x+3)^2 + (y-4)^2 = 25 \]
This represents a circle centered at \( (-3, 4) \) with a radius of \( 5 \).

### Step 2: Find the intersection points of the line with the circles
Substituting \( y = mx \) into the equations of the circles:

For the first circle:
\[ x^2 + (mx)^2 - 2x - 2(mx) = 0 \implies (1 + m^2)x^2 - (2 + 2m)x = 0 \]
Factoring out \( x \):
\[ x \left( (1 + m^2)x - (2 + 2m) \right) = 0 \]
The solutions are \( x = 0 \) (the origin) and \( x = \frac{2 + 2m}{1 + m^2} \).

For the second circle:
\[ x^2 + 6x + (mx)^2 - 8(mx) = 0 \implies (1 + m^2)x^2 + (6 - 8m)x = 0 \]
Factoring out \( x \):
\[ x \left( (1 + m^2)x + (6 - 8m) \right) = 0 \]
The solutions are \( x = 0 \) (the origin) and \( x = \frac{8m - 6}{1 + m^2} \).

### Step 3: Determine the conditions for the origin to divide \( AB \) internally
The points \( A \) and \( B \) are given by:
- \( A = \left( \frac{2 + 2m}{1 + m^2}, m \cdot \frac{2 + 2m}{1 + m^2} \right) \)
- \( B = \left( \frac{8m - 6}{1 + m^2}, m \cdot \frac{8m - 6}{1 + m^2} \right) \)

For the origin to divide \( AB \) internally, the x-coordinates of points \( A \) and \( B \) must have the same sign.

### Step 4: Analyze the conditions
1. For \( A \):
\[ \frac{2 + 2m}{1 + m^2} > 0 \implies 2 + 2m > 0 \implies m > -1 \]

2. For \( B \):
\[ \frac{8m - 6}{1 + m^2} > 0 \implies 8m - 6 > 0 \implies m > \frac{3}{4} \]

### Step 5: Combine the conditions
The combined conditions for \( m \) are:
- \( m > -1 \)
- \( m > \frac{3}{4} \)

Thus, the range of \( m \) such that the origin divides \( AB \) internally is:
\[ m > \frac{3}{4} \]

### Final Answer
The range of \( m \) such that the origin divides \( AB \) internally is \( m > \frac{3}{4} \).