The area of the triangle formed by the line x+y-8=0 and the pair of straight lines through origi

step 1 So in this problem we have been given the area of the triangle formed by the line x plus y minus

The area of the triangle formed by the line x+y-8=0 and the...

The area of the triangle formed by the line $\mathrm{x}+\mathrm{y}-8=0$ and the pair of straight lines through origin, which pass through the intersection of $x^{2}+y^{2}-2 x-2 y-2=0$ and $x+y-4=0$
(a) 32
(b) 8
(c) 16
(d) 24

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Key Concepts

Intersection of a Circle and a Line

This concept involves finding the common points where a circle and a straight line meet. It typically requires substituting the linear equation into the circle’s equation to obtain a quadratic equation, whose solutions yield the coordinates of the intersection points. This technique is fundamental in coordinate geometry, as it applies to problems where curves and lines are combined.

Pair of Lines Through a Fixed Point

This idea refers to the family of lines that all pass through a given point, in this case, the origin. When additional conditions—such as passing through one or more specific points—are imposed, it results in a unique pair of lines. Understanding how to determine the equations of these lines is essential in problems that involve the construction of figures from multiple intersecting lines.

Triangle Area Calculation in Coordinate Geometry

This concept addresses finding the area of a triangle when the coordinates of its vertices are known or can be determined from intersections of lines. Techniques often involve using the determinant formula or base-height methods after finding the intersection points among lines. It is a key application of coordinate geometry that requires combining knowledge of slopes, intercepts, and algebraic manipulation.

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