Explore advanced tenets of coordinate geometry (angle between two lines, perpendicular distance, locus of a moving point, ratio theorem, and equations of a circle)

The shortest distance from a point P(x₁, y₁) to the line ax + by + c = 0 is:

$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$

Worked Example 2

Find the perpendicular distance from the point (1, 3) to the line 2x + 4y + 3 = 0.

Solution

Here a = 2, b = 4, c = 3, and (x₁, y₁) = (1, 3)

$d = \frac{|2(1) + 4(3) + 3|}{\sqrt{2^2 + 4^2}} = \frac{|2 + 12 + 3|}{\sqrt{20}} = \frac{17}{\sqrt{20}} = 3.8 \text{ units}$

A locus is the path traced by a point that moves according to a given condition.

Worked Example 3

Find the equation of the locus of a point P(x, y) that is equidistant from points (2, 4) and (0, -3).

Solution

Using the distance formula: $\sqrt{(x-2)^2 + (y-4)^2} = \sqrt{(x-0)^2 + (y+3)^2}$

Squaring both sides: $(x-2)^2 + (y-4)^2 = x^2 + (y+3)^2$

Expanding: $x^2 - 4x + 4 + y^2 - 8y + 16 = x^2 + y^2 + 6y + 9$

Simplifying: $-4x - 14y + 11 = 0$

Therefore, the locus is the straight line: 4x + 14y - 11 = 0

Internal Division

A point P(x, y) dividing the line joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n has coordinates:

$P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$

External Division

For external division in ratio m:n:

$P = \left(\frac{nx_1 - mx_2}{n-m}, \frac{ny_1 - my_2}{n-m}\right)$

Worked Example 4

Find the coordinates of the point that divides internally the line joining A(-4, 10) and B(8, 4) in the ratio 1:3.

Solution

Given m = 1, n = 3, A(-4, 10), B(8, 4)

$P = \left(\frac{1(8) + 3(-4)}{1+3}, \frac{1(4) + 3(10)}{1+3}\right) = \left(\frac{8-12}{4}, \frac{4+30}{4}\right) = \left(-1, \frac{17}{2}\right)$

A circle is the locus of a point that is at a constant distance (radius) from a fixed point (centre).

Standard Form

For a circle with centre (a, b) and radius r:

$(x-a)^2 + (y-b)^2 = r^2$

General Form

$x^2 + y^2 + 2gx + 2fy + c = 0$

With centre (-g, -f) and radius = √(g² + f² - c)

Worked Example 5

Find the centre and radius of the circle x² + y² - 4x + 2y + 1 = 0.

Solution

Comparing with the general form: g = -2, f = 1, c = 1

Centre = (-g, -f) = (2, -1)

Radius = √((-2)² + 1² - 1) = √(4 + 1 - 1) = √4 = 2 units

Alternatively, complete the square: $x^2 - 4x + 4 + y^2 + 2y + 1 = 4$ $(x-2)^2 + (y+1)^2 = 2^2$

In Tanzania, coordinate geometry is used in land surveying and map reading. For example, when a farmer wants to determine the exact area of a circular fish pond or when a town planner needs to calculate the distance of a proposed market site from a main road, these coordinate geometry formulas help determine distances, boundaries, and appropriate locations efficiently.