Introduction to the Problem

This article discusses how to find the area of triangle POQ when points P and Q lie on a given line and are at a distance of 5 units from the origin. We will use both algebraic and geometric approaches to solve this problem, leveraging properties of lines, circles, and the concept of perpendicular lines.

Underlying Equations and Key Concepts

Let's start with the given line and circle equations:

The line: (text{x - y -1}) The circle: ({text{x}^2 text{y}^2 25})

The goal is to find points P and Q on the line that are 5 units away from the origin and then calculate the area of triangle POQ, where O is the origin (0,0).

Step-by-Step Solution

Step 1: Convert the Line Equation to Slope-Intercept Form

The given line equation can be rewritten in the form (text{y x 1}).

Why: Subtract (text{x}) from both sides of the equation.

Step 2: Find Points of Intersection of the Line and Circle

Substitute (text{y x 1}) into the circle equation ({text{x}^2 text{y}^2 25}).

Algebraic Solution:

({text{x}^2 (text{x} 1)^2 25})

({text{x}^2 text{x}^2 2text{x} 1 25})

({2text{x}^2 2text{x} - 24 0})

({text{x}^2 text{x} - 12 0})

This quadratic equation can be factored:

({(text{x} - 3)(text{x} 4) 0})

Thus, (text{x 3}) or (text{x -4}).

For (text{x 3}), (text{y x 1 4}).

For (text{x -4}), (text{y x 1 -3}).

Therefore, the points P and Q are (3, 4) and (-4, -3).

Step 3: Calculating the Length of the Base (Segment PQ)

The distance between points P (3, 4) and Q (-4, -3) is calculated using the distance formula.

The distance (d sqrt{(3 - (-4))^2 (4 - (-3))^2} sqrt{7^2 7^2} sqrt{98} 7sqrt{2}).

Step 4: Finding the Altitude from the Origin to Line Segment PQ

The slope of the line segment PQ is 1 (since the line equation is y x 1).

The slope of the perpendicular line through the origin is -1 (negative reciprocal).

The equation of the line perpendicular to PQ through the origin is: (text{y -x}).

Find the intersection of y -x and y x 1.

(-text{x} text{x} 1 Rightarrow 2text{x} -1 Rightarrow text{x} -frac{1}{2}, text{y} frac{1}{2}).

The altitude is the distance from the origin (0,0) to the point ((- frac{1}{2}, frac{1}{2})), which is (sqrt{left(frac{1}{2}right)^2 left(frac{1}{2}right)^2} sqrt{frac{1}{4} frac{1}{4}} sqrt{frac{1}{2}}).

Step 5: Calculating the Area of Triangle POQ

The area of a triangle is given by (frac{1}{2} times text{base} times text{height}).

(text{Base} 7sqrt{2}), (text{Height} sqrt{frac{1}{2}}).

(text{Area} frac{1}{2} times 7sqrt{2} times sqrt{frac{1}{2}} frac{1}{2} times 7sqrt{2} times frac{sqrt{2}}{2} frac{1}{2} times 7 frac{7}{2}).

Conclusion and Summary

We have shown that the area of triangle POQ, given the line x - y -1 and points at a distance of 5 units from the origin, is (frac{7}{2}).

Additional Insights

Using the concept of the circle and quadratic equations, we can explore various geometric and algebraic properties of lines and distances.