Coordinate Geometry:

A locus is simply the path traced by a point that satisfies a particular condition. When we talk about loci relating to parallel lines, we're finding all points that maintain equal distance from a given line or between two parallel lines.

Imagine you're walking along Lekki-Epe Expressway in Lagos, keeping exactly the same distance from the road at all times. Your path would form a line parallel to the expressway—that's your locus. If you have two parallel roads, any point equidistant from both roads lies on a straight line running exactly between them.

In coordinate geometry, if you have a line and need to find all points at distance d from it, these points form two parallel lines, one on each side of the original line. This concept appears frequently in JAMB questions involving distances and parallel conditions.

In coordinate geometry, understanding lines is fundamental. A line connects two points and can be described using equations like y = mx + c, where m is the slope. A perpendicular bisector is a special line that cuts another line segment exactly in half at a right angle (90 degrees). Think of it like drawing a perfectly straight road through the middle of Lagos Island, perpendicular to the Lagoon's edge.

An angle bisector divides an angle into two equal parts. When solving JAMB questions, you'll need to find equations of these lines using coordinates. The key skill involves calculating midpoints, slopes, and using the perpendicular relationship where slopes multiply to give -1.

These concepts frequently appear together in coordinate geometry problems, testing your ability to manipulate equations and understand geometric properties algebraically.

Imagine two locations on Lagos Island—one at point A and another at point B. The midpoint is simply the exact centre between them. To find it, add the x-coordinates together and divide by 2, then do the same for the y-coordinates. So if A is at (2, 3) and B is at (8, 7), your midpoint is ((2+8)/2, (3+7)/2) = (5, 5).

The gradient tells you how steep a line is—basically how much the line rises or falls as you move along it. It's the change in y divided by the change in x. Between those same two points, the gradient is (7-3)/(8-2) = 4/6 = 2/3. A positive gradient means the line slopes upward; a negative gradient means it slopes downward.

The distance between two points on a coordinate plane is simply how far apart they are. Imagine two locations on a map of Lagos—say the National Museum at coordinates (2, 3) and Lekki Conservation Centre at (5, 7). To find how far apart these places are, we use the distance formula: the square root of (x₂ - x₁)² + (y₂ - y₁)².

So the distance becomes: √[(5-2)² + (7-3)²] = √[9 + 16] = √25 = 5 units. This formula comes from the Pythagorean theorem because the two points form a right triangle when you draw them on a graph.

The key is identifying your coordinates correctly and substituting them carefully into the formula. Most students make mistakes when subtracting negative numbers, so pay close attention to your signs.

Two lines are perpendicular when they meet at a right angle, forming an L-shape. The key to identifying perpendicular lines mathematically is their gradients (slopes). When two lines are perpendicular, the product of their gradients equals negative one. So if one line has gradient m₁ and another has gradient m₂, then m₁ × m₂ = -1.

Think of the roads around Lagos Island. If one street runs north-south and another runs east-west, they're perpendicular. To check this mathematically, find each road's gradient from their equations. If their gradients multiply to give -1, they're definitely perpendicular.

This concept appears constantly in JAMB questions about finding equations of perpendicular lines or proving perpendicularity between two given lines.

When you know two points on a straight line, you can find its equation using the two-point formula. The formula is: (y - y₁)/(x - x₁) = (y₂ - y₁)/(x₂ - x₁). Think of it this way: you're finding the slope between your two points, then using it to write the complete equation.

Imagine a straight road connecting Lagos and Ibadan on a coordinate map where Lagos is at point (2, 3) and Ibadan is at (5, 9). To find the equation of this road, you'd substitute these coordinates into the formula. The slope becomes (9 - 3)/(5 - 2) = 2. Using point-slope form, your equation becomes y - 3 = 2(x - 2), which simplifies to y = 2x - 1.