Intersection of a Line and Curve
Intersection of a Line and Curve
TGM Original Questions
- A simultaneous system is non-linear when at least one of its equations is non-linear.
- In this unit it will involve quadratic equations of the general form \(y = ax^2 + bx + c\) and a linear equation with a general form of \(y = mx + d\).
- The solution of the simultaneous equation will be the point(s) of intersection between the line and the curve.
- The nature of the point(s) of intersections, can be used to find unknown coefficients in the equation and is summarised in the table below:
Nature of points of intersection | Graphical representation | Discriminant |
|---|---|---|
Line intersects with the curve at \(2\) points | \(B^2 - 4AC > 0\) | |
The line intersects is tangent to the curve and intersects at \(1\) point | \(B^2 - 4AC = 0\) | |
The line does not intersect the curve | \(B^2 - 4AC < 0\) |
After substitution, arrange the intersection equation as \(Ax^2+Bx+C=0\), where \(A\ne0\). The table uses its discriminant \(B^2-4AC\), not the coefficients of the curve before substitution.
For the displayed curve and line, \(A=a\), \(B=b-m\) and \(C=c-d\).
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