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The concept of intercept length in coordinate geometry refers to the distance between two points where a curve or line cuts an axis or another line. When the curve, such as a circle described by the equation \(x^{2}+y^{2}+2ex+2fy+c=0\), intersects a line like \(px+qy=1\), the segment of the line that lies inside the curve is called the intercept.

To determine the intercept length, you generally need to find the points of intersection between the curve and the line. This involves solving the equations of the curve and line simultaneously. Once the intersection points are determined, you can calculate their distance apart, which gives you the length of the intercept.

In the specific case provided, the length of the intercept is unique because it subtends a right angle at the origin, leading to some interesting properties that relate directly to the condition for perpendicularity, thus allowing us to develop a specific mathematical relationship as shown in the exercise.

The condition for perpendicularity is a fundamental concept in coordinate geometry, particularly when it comes to understanding the relationships between two intersecting lines or segments. Two lines are perpendicular to each other if the angle between them is 90 degrees. In terms of slope, if the slope of one line is \(m_1\) and the other is \(m_2\), then the lines are perpendicular if and only if \(m_1 \times m_2 = -1\).

This concept is extended to any two vectors in a plane; the vectors are perpendicular if their dot product equals zero. In the context of our exercise, the vectors from the origin to each point of intersection with the line create a right angle, meaning their dot product should be zero. This dot product condition translates algebraically into the equation that we ultimately want to prove.

Application in the Exercise

During the exercise, we applied this concept by setting up an equation that relates the coordinates of the points of intersection and setting their dot product to zero. This step is vital as it mathematically encodes the right angle subtended by the intercept length at the origin, allowing us to derive the required expression.

Understanding quadratic equations is crucial in various coordinate geometry problems, including those involving curves like circles, parabolas, and ellipses. A quadratic equation follows the standard form \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The solutions to quadratic equations are the x-values where the parabola, which is the graph of the quadratic equation, intersects the x-axis.

To find these intersecting points, one must apply methods like factoring, completing the square, or using the quadratic formula. In the context of the original exercise, once the x-coordinate from the line equation is substituted into the curve equation, a quadratic in terms of y is obtained. Solving this quadratic yields the y-coordinates of the points where the line and the curve intersect.

Significance in the Exercise

Solving the quadratic equation is a pivotal step in the exercise because the solutions provide the exact points where the line meets the curve. These points are necessary to determine the length of the intercept and to apply the condition for perpendicularity, which together lead to the final expression we aim to show.