Find the equation of the straight line passing through the origin and the point of intersection of the lines x/a + y/b = 1 and x/b + y/a = 1.

Options

A. y = x
B. y = -x
C. y = 2x
D. y = -2x

Answer

To find the equation of the straight line passing through the origin and the point of intersection of the lines given by the equations \( \frac{x}{a} + \frac{y}{b} = 1 \) and \( \frac{x}{b} + \frac{y}{a} = 1 \), we will follow these steps:

### Step 1: Rewrite the equations in standard form
The given equations can be rewritten as:
1. \( \frac{x}{a} + \frac{y}{b} = 1 \) can be multiplied by \( ab \):
\[ b x + a y = ab \]
2. \( \frac{x}{b} + \frac{y}{a} = 1 \) can also be multiplied by \( ab \):
\[ a x + b y = ab \]

### Step 2: Set up the equations for elimination
Now we have the two equations:
1. \( b x + a y = ab \) (Equation 1)
2. \( a x + b y = ab \) (Equation 2)

### Step 3: Eliminate one variable
To find the point of intersection, we can eliminate \( y \) by making the coefficients of \( y \) the same. We can multiply Equation 1 by \( b \) and Equation 2 by \( a \):
1. \( b^2 x + a b y = ab^2 \)
2. \( a^2 x + b^2 y = a b^2 \)

### Step 4: Subtract the equations
Now we subtract the modified Equation 2 from modified Equation 1:
\[ (b^2 x + a b y) - (a^2 x + b^2 y) = ab^2 - ab^2 \]
This simplifies to:
\[ (b^2 - a^2)x + (ab - b^2)y = 0 \]

### Step 5: Solve for \( x \)
Rearranging gives:
\[ (b^2 - a^2)x = (b^2 - ab)y \]
We can express \( x \) in terms of \( y \):
\[ x = \frac{(b^2 - ab)}{(b^2 - a^2)}y \]

### Step 6: Substitute to find \( y \)
Now we can substitute \( x \) back into one of the original equations to find \( y \). Let's use Equation 1:
\[ b\left(\frac{(b^2 - ab)}{(b^2 - a^2)}y\right) + ay = ab \]
This simplifies to:
\[ \frac{b(b^2 - ab)}{(b^2 - a^2)}y + ay = ab \]
Multiply through by \( (b^2 - a^2) \) to eliminate the fraction:
\[ b(b^2 - ab)y + ay(b^2 - a^2) = ab(b^2 - a^2) \]

### Step 7: Solve for \( y \)
Combine like terms and solve for \( y \):
\[ (b^2 - ab + a(b^2 - a^2))y = ab(b^2 - a^2) \]
Thus, we can find \( y \) and then substitute back to find \( x \).

### Step 8: Find the slope of the line
Now we have the coordinates of the intersection point. The slope \( m \) of the line passing through the origin (0,0) and the intersection point \((x_0, y_0)\) is given by:
\[ m = \frac{y_0 - 0}{x_0 - 0} = \frac{y_0}{x_0} \]

### Step 9: Write the equation of the line
The equation of the line in slope-intercept form is:
\[ y = mx \]

### Final Step: Substitute the slope
Substituting the value of \( m \) we found into the equation gives us the final equation of the line.