91Ó°ÊÓ

The concept of the perpendicular foot involves finding the point on a line where a perpendicular segment from a given point, often the origin (0,0), meets the line. It is a very common concept in coordinate geometry and helps in solving various geometric problems.
To determine the perpendicular foot, we first need to find the slope of the given line. For a line like \(3x + y = \lambda\), rewrite it in slope-intercept form, \(y = -3x + \lambda\), showing that the slope is \(-3\).
The perpendicular line from the origin will have a slope that is the negative reciprocal of the original line. In this case, the perpendicular slope is \(-\frac{1}{3}\), because multiplying \(-3\) by \(-\frac{1}{3}\) results in a slope of \(1\), the condition for perpendicular lines.
The equation of the line with this slope passing through the origin (point \((0,0)\)) is \(y = -\frac{1}{3}x\). Solving this together with the original line equation, \(3x + y = \lambda\), provides us the coordinates of the perpendicular foot \(P\). This intersection gives a systematic point \(P\), in our example \(\left( \frac{3\lambda}{10}, -\frac{\lambda}{10} \right)\).

Intercepts are the points where a line crosses the coordinate axes, namely the x-axis and y-axis. Understanding intercepts is crucial in coordinate geometry, as it helps describe the line's position and shape.
Finding the x-intercept involves setting \(y = 0\) in the line equation. From the line equation \(3x + y = \lambda\), by setting \(y\) to zero, we can solve for \(x\) and find the x-intercept, \(A\), at \(\left( \frac{\lambda}{3}, 0 \right)\).
Similarly, finding the y-intercept involves setting \(x = 0\). By substituting \(x = 0\) into the equation \(3x + y = \lambda\), we find \(y = \lambda\), giving the y-intercept, \(B\), at \((0, \lambda)\).
These intercepts provide significant points where the line crosses the axes, forming the basis for further geometric constructions, such as drawing a triangle and determining intercepts related ratios.

The distance formula is a fundamental tool in coordinate geometry used to calculate the distance between two points in Cartesian coordinates. The formula is rooted in the Pythagorean theorem and is given by the expression:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points in question.
To find distances such as between points \(P\left( \frac{3\lambda}{10}, -\frac{\lambda}{10} \right)\) and \(A\left( \frac{\lambda}{3}, 0 \right)\), we substitute into the formula. Calculate the differences in x-values and y-values, square these differences, sum them, and take the square root. For these points specifically, this results in \(PA = \frac{\lambda}{6}\).
For another pair, such as \(P\left( \frac{3\lambda}{10}, -\frac{\lambda}{10} \right)\) and \(B(0, \lambda)\), follow the same method to find the distance \(PB = \frac{\lambda}{2}\). These calculations allow us to compare distances, crucial for problems involving ratios or optimizations.