Points, segments and lines: ratios, distances and loci · Year 12
In Points, segments and lines: ratios, distances and loci (Year 12) you practise 2 skills: The shortest distance · Loci that are lines. A point that divides a segment in a given ratio lies on the segment, and its distances from the two ends are in that ratio. If the ratio is two to three, the segment splits into five equal parts and the point is two of them along: to each coordinate of the first end, add two fifths of the difference between the ends. The distance between two objects is the least distance between a point of one and a point of the other. The distance from a point to a line is measured along the perpendicular, and the distance from a point outside a circle to the circle is its distance from the centre minus the radius.
There is a formula for the distance from a point to a line: write the line’s equation with zero on one side, put the point’s coordinates into the other side, take the absolute value and divide by the square root of the sum of the squares of the two coefficients. For the distance between two parallel lines, take a point on one and find its distance from the other, or write both equations with the same coefficients and divide the difference of the constant terms, as an absolute value, by that square root. A locus is the set of all points that meet a condition. Some loci are straight lines. The points the same distance from two points lie on the perpendicular bisector of the segment between them, the points the same distance from two parallel lines lie on the parallel line midway between them, and the points the same distance from two lines that meet lie on the two bisectors of the angles between them. The third vertex of a triangle with a fixed base and a fixed area also moves on two lines parallel to the base. A model proof that the points the same distance from the two ends of a segment form its perpendicular bisector: choose the axes so the segment lies on the horizontal axis with its midpoint at the origin, so the horizontal coordinates of its ends are a number and its negative. The square of the distance from a point to each end is the square of the difference in the horizontal coordinate plus the square of the vertical coordinate. Setting the two squares equal and expanding, everything cancels except the term in which the point’s horizontal coordinate is multiplied by the end’s horizontal coordinate, and it appears on the two sides with opposite signs. So the point’s horizontal coordinate is zero, and every such point is on the vertical axis. The other way round, every point on the vertical axis is the same distance from the two ends. The vertical axis passes through the midpoint of the segment at right angles to it, so it is the perpendicular bisector.
Tip: In a locus problem, label a general point by its coordinates, write the condition as an equation and simplify it.
What you learn
The shortest distance
For example: The line x − y = 11 and the point P(5, −4) are given. What are the coordinates of the point on the line nearest to P ?
Answer and solution
The answer: (6, −5)
The perpendicular to the line through P is x + y = 1 . Solving it together with the line gives x = 6 and y = −5 , so the point is (6, −5) .
Loci that are lines
For example:
The points A(7, 4) and B(−5, −6) are given. What is the locus of all points the same distance from A and from B ?- 5x − 6y = 11
- 6x + 5y = 62
- 6x − 5y = 11
- 6x + 5y = 1
Answer and solution
The answer: 6x + 5y = 1
The midpoint is (1, −1) , and the coordinates change by (−12, −10) from A to B . The line 6x + 5y = 1 passes through the midpoint, and it is perpendicular to the segment since its coefficients of x and y are in the same ratio as −12 and −10 .
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).