Testing coordinate incidence
The residual is the value of the left side of a line equation after a point is substituted. A zero residual means the point is on the line, while a non-zero residual means it is not
\[r=Ax+By+C\]
- Substitute both coordinates Replace x and y in the same equation with the point's coordinates \(\displaystyle r=2(3)-4-2\)
- Use a zero residual A residual of zero verifies that the point lies on the line \(\displaystyle r=2(3)-4-2=0\Rightarrow(3,4)\text{ is on }2x-y-2=0\)
- Interpret a non-zero residual A non-zero result rejects the point for that line \(\displaystyle r=2(3)-5-2=-1\ne0\)
- Check every required item Collinearity and concurrency require the remaining point or line to satisfy the same condition \(\displaystyle I=g_1\cap g_2,\;r_{g_3}(I)=0\)
Worked example
Determine whether the given point lies on the given line
\[P(4,5),;2x-y-3=0,;r=2(4)-5-3\]
- \(\displaystyle 2(4)=8\)Substitute the x-coordinate into the 2x term
- \(\displaystyle 8-5=3\)Subtract the y-coordinate from the partial result
- \(\displaystyle 3-3=0\)Include the constant term to obtain the residual
- \(\displaystyle r=0\Rightarrow P(4,5)\text{ lies on the line}\)A zero residual verifies incidence
Common mistakes
Checking only one coordinate
✕\(\displaystyle 2(3)-2=4\Rightarrow(3,4)\text{ is not on the line}\)✓\(\displaystyle 2(3)-4-2=0\Rightarrow(3,4)\text{ is on the line}\)
Incidence requires substituting both coordinates and the constant into the same equationReversing the position residual
✕\(\displaystyle r=2(3)-5-2=-1\Rightarrow\text{below}\)✓\(\displaystyle r=-1,\;B=-1\Rightarrow r/B=1>0\Rightarrow\text{above}\)
For a non-vertical line, the residual must be interpreted using the coefficient of y to determine above or belowAssuming pairwise intersections are concurrent
✕\(\displaystyle g_1\cap g_2=I,\;g_2\cap g_3=J\Rightarrow\text{concurrent}\)✓\(\displaystyle I=J\;\text{and}\;I\in g_3\Rightarrow\text{concurrent}\)
All lines must contain the same point, not merely have intersections in pairsDifficulty levels
introClassify a point as on, above or below a nonvertical line, or on, left or right of a vertical line
standardTest whether three points are collinear
challengeTest concurrency or determine a parameter for incidence or parallelism
examChoose a coordinate verdict or solve a compact parameter condition
Questions about this topic
What is a residual in a line equation?+
It is the value of the left side of the line equation after the coordinates of a point are substituted
How is collinearity tested?+
Use two points to form a line, then substitute every remaining point into that line equation and check for zero residuals
Are pairwise intersections enough to prove concurrency?+
No. The same coordinate point must lie on every line in the set
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 1: Solving equations
In this chapter
Factorise, solve and reconstruct a rational quadraticSolve a quadratic exactly with the quadratic formulaSolve an equation that reduces to quadratic formDetermine parameter values from a discriminant conditionDivide polynomials and state quotient and remainderApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsDetermine parameters in a polynomial identityUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityUse exact trigonometric identities and quadrant informationSolve circular equations over finite domainsGeneral solutions of circular equations and interval extractionLocate and bisect a circular-equation rootRearrange literal equations and recognise equivalent line formsSolve a pair of simultaneous linear equationsSolve a linear system with three or four unknownsVerify coordinate incidence and determine a parameterSolve literal equations and simultaneous literal equationsClassify and parameterise linear systems