Find the equations to the straight lines passing through the following pairs of points.
`(a cos phi_(1), b sin phi_(1)) and (a cos phi_(2), b sin phi_(2))`

Answer

To find the equation of the straight line passing through the points \((a \cos \phi_1, b \sin \phi_1)\) and \((a \cos \phi_2, b \sin \phi_2)\), we can use the two-point form of the equation of a line. Here are the steps to derive the equation:

### Step 1: Identify the Points
Let the points be:
- Point 1: \(P_1 = (x_1, y_1) = (a \cos \phi_1, b \sin \phi_1)\)
- Point 2: \(P_2 = (x_2, y_2) = (a \cos \phi_2, b \sin \phi_2)\)

### Step 2: Use the Two-Point Form of the Line Equation
The two-point form of the equation of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]
Substituting the coordinates of the points:
\[ y - b \sin \phi_1 = \frac{b \sin \phi_2 - b \sin \phi_1}{a \cos \phi_2 - a \cos \phi_1} (x - a \cos \phi_1) \]

### Step 3: Simplify the Slope
The slope of the line can be simplified:
\[ \text{slope} = \frac{b (\sin \phi_2 - \sin \phi_1)}{a (\cos \phi_2 - \cos \phi_1)} \]
Thus, the equation becomes:
\[ y - b \sin \phi_1 = \frac{b (\sin \phi_2 - \sin \phi_1)}{a (\cos \phi_2 - \cos \phi_1)} (x - a \cos \phi_1) \]

### Step 4: Rearranging the Equation
Multiplying both sides by \(a (\cos \phi_2 - \cos \phi_1)\) to eliminate the fraction:
\[ a (\cos \phi_2 - \cos \phi_1)(y - b \sin \phi_1) = b (\sin \phi_2 - \sin \phi_1)(x - a \cos \phi_1) \]

### Step 5: Expand and Rearrange
Expanding both sides:
\[ a (\cos \phi_2 - \cos \phi_1) y - ab (\cos \phi_2 - \cos \phi_1) \sin \phi_1 = b (\sin \phi_2 - \sin \phi_1) x - ab (\sin \phi_2 - \sin \phi_1) \cos \phi_1 \]
Rearranging gives us:
\[ b (\sin \phi_2 - \sin \phi_1) x + a (\cos \phi_2 - \cos \phi_1) y = ab (\sin \phi_2 - \sin \phi_1) \cos \phi_1 + ab (\cos \phi_2 - \cos \phi_1) \sin \phi_1 \]

### Step 6: Final Equation
Thus, the final equation of the line is:
\[ b (\sin \phi_2 - \sin \phi_1) x + a (\cos \phi_2 - \cos \phi_1) y = ab (\sin \phi_2 \cos \phi_1 + \cos \phi_2 \sin \phi_1 - \sin \phi_1 \cos \phi_1) \]