A locus is a set of points traced out by following a set of instructions. The instructions describe how one point is built from another, and the locus is the path that point sweeps out as the construction is repeated.

True or False? The locus of a point constructed from a conic can be a completely different type of curve from the conic it was built on.

True. On a rectangular hyperbola the intersection of two tangents can trace a straight line , while on an ellipse the midpoints of certain line segments trace a different ellipse . It is the construction, not the original curve, that decides what the locus turns out to be.

A LevelFurther MathsEdexcelFurther Pure 1FlashcardsCoordinate SystemsLoci Problems

Loci Problems (Edexcel A Level Further Maths: Further Pure 1): Flashcards

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FrontLoci Problems
Define locus.

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Cards in this collection (7)

Define locus.
A locus is a set of points traced out by following a set of instructions.
The instructions describe how one point is built from another, and the locus is the path that point sweeps out as the construction is repeated.
A question asks for the locus of a point built from a construction on a conic. What must the coordinates of that point be written in terms of before anything else can be done?
They must be written in terms of the parameter of the conic, $t$ or $θ$ , the single letter that both coordinates depend on.
Only then can the parameter be eliminated, which is the step that turns a moving point into an equation.
Fill in the two words describing what each stage of a locus calculation produces:
Setting $X$ and $Y$ equal to the coordinates of the moving point gives the $______$ equations of the locus, and eliminating the parameter then gives its $______$ equation.
The completed sentence is:
Setting $X$ and $Y$ equal to the coordinates of the moving point gives the parametric equations of the locus, and eliminating the parameter then gives its Cartesian equation.
The parametric pair also shows the range of values $X$ and $Y$ can take, which the Cartesian equation on its own does not.
When finding a locus, why is it worth using capital $X$ and $Y$ for the moving point's coordinates rather than $x$ and $y$ ?
Because $x$ and $y$ are already in use in the equation of the curve and in the equations of its tangents and normals, so reusing them invites a collision in the working.
The capitals keep the locus separate until the parameter has gone, and the final answer is then written back in lower case.
True or False?
The locus of a point constructed from a conic can be a completely different type of curve from the conic it was built on.
True.
On a rectangular hyperbola the intersection of two tangents can trace a straight line, while on an ellipse the midpoints of certain line segments trace a different ellipse.
It is the construction, not the original curve, that decides what the locus turns out to be.
A straight line through the point $P$ on a parabola cuts the curve again at $Q$ , and solving the two equations together gives a quadratic in $x$ . How do you use the fact that $P$ is already known?
The $x$ -coordinate of $P$ is already one root of that quadratic, so the bracket it comes from must be a factor.
Dividing that factor out leaves a linear bracket, and solving it gives the $x$ -coordinate of $Q$ without any further quadratic work.
You have the $x$ -coordinate of $Q$ , where a straight line meets a parabola for the second time. Why find its $y$ -coordinate from the line rather than from the parabola?
Because the parabola's equation is quadratic in $y$ , so it returns $y = \pm \dots$ and leaves you to work out which sign belongs to $Q$ .
The equation of the line gives $y$ directly, with no ambiguity to resolve.

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