A straight line passing through the point (87, 33) cuts the positive direction of the coordinate axes at the point P and Q. If Q is the origin then the minimum area of the triangle OPQ is.

Answer

To solve the problem step by step, we need to find the minimum area of triangle OPQ where O is the origin, P is the point where the line intersects the y-axis, and Q is the point where the line intersects the x-axis. The line passes through the point (87, 33) and Q is the origin (0, 0).

### Step 1: Equation of the Line
The line passing through the point (87, 33) can be expressed in slope-intercept form. Let the slope of the line be \( m \).

Using the point-slope form of the line:
\[ y - 33 = m(x - 87) \]
Rearranging gives us:
\[ y = mx - 87m + 33 \]

### Step 2: Finding Intercepts
To find the intercepts, we need to determine where the line intersects the x-axis (Q) and the y-axis (P).

**Finding Q (x-intercept)**:
Set \( y = 0 \):
\[ 0 = mx - 87m + 33 \]
Solving for \( x \):
\[ mx = 87m - 33 \implies x = \frac{87m - 33}{m} \]
Thus, the coordinates of Q are:
\[ Q\left(\frac{87m - 33}{m}, 0\right) \]

**Finding P (y-intercept)**:
Set \( x = 0 \):
\[ y = m(0) - 87m + 33 = -87m + 33 \]
Thus, the coordinates of P are:
\[ P(0, -87m + 33) \]

### Step 3: Area of Triangle OPQ
The area \( A \) of triangle OPQ can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the x-coordinate of Q and the height is the y-coordinate of P:
\[ A = \frac{1}{2} \times \frac{87m - 33}{m} \times (-87m + 33) \]

### Step 4: Simplifying the Area Expression
Substituting the values:
\[ A = \frac{1}{2} \times \frac{(87m - 33)(-87m + 33)}{m} \]
Expanding the product:
\[ A = \frac{1}{2} \times \frac{-87m \cdot 87m + 33 \cdot 87m + 33 \cdot 33 - 33 \cdot 87m}{m} \]
This simplifies to:
\[ A = \frac{1}{2} \times \frac{-7569m^2 + 1089}{m} \]

### Step 5: Finding the Minimum Area
To find the minimum area, we differentiate \( A \) with respect to \( m \) and set the derivative to zero:
\[ \frac{dA}{dm} = 0 \]
Calculating the derivative gives:
\[ \frac{dA}{dm} = \frac{1}{2} \left( \frac{-15138m + 1089}{m^2} \right) = 0 \]
Solving for \( m \):
\[ -15138m + 1089 = 0 \implies m = \frac{1089}{15138} \]

### Step 6: Substitute Back to Find Area
Substituting \( m = \frac{33}{87} \) back into the area formula:
\[ A = \frac{1}{2} \times \frac{87 \cdot 33}{2} \]
Calculating gives:
\[ A = \frac{33 \cdot 87}{2} \]

### Final Calculation
Calculating the final area:
\[ A = \frac{33 \cdot 87}{2} = \frac{2871}{2} = 1435.5 \]

Thus, the minimum area of triangle OPQ is:
\[ \text{Minimum Area} = 1435.5 \]