Locus Flashcards
Simple locus circle
A circle is the locus of all points a fixed distance (radius) from a point (centre)
Sample Question:
Find the equation of the locus of a point P(x,y) that moves so that it is always 5 units away from (1,0)
A locus is:
a) set of points which satisfy a given condition (x,y)
b) point P(x,y) that moves along a certain path
A locus is an equation which describes the points
Simple locus SLG
A SLG is the locus of points equidistant from two points or lines.
Simple locus parabola
The locus of all points equidistant from a point and a line
Parabolas locus in basic form
x^2= +/- 4ay
Focus=
F(0,a)
Latus rectum=
4a
Directrix=
y= +/- a
Parabolas in inverse basic form
y^2= +/- 4ax
Directrix in inverse form=
x=+/- a
Inverse focus=
F(a,0)
Parabolas in general form
(x-A)^2=4a(y-B) where the vertex is (A,B)
General form focus=
F(A,B+a)
General form directrix=
y=B-a
Parabolas in inverse general form
(y-B)^2= 4a(x-A)
IMPORTANT
Notice how not only the x and y values have replaced each other, but the whole (y-B) and (x-A) terms.
Parabolas in inverse general form focus=
F(A+a, B)
Parabolas in inverse general form directrix=
x=A-a
How to convert a parabola equation in general form locus to find its vertex:
1) locate the term with the highest power
2) move all terms that do not contain the term with the highest power to the RHS
E.g. x^2-6x-16y=41 —> x^2-6x=16y+41
3) complete the square for the LHS
4) factorise the RHS and change it into +/-4a(y-B) form
Parametric Equations
–> equations in which two parameters are expressed in terms of a third parameter i.e. X and y in terms of t
Parametric equations of the parabola:
–>the cartesian equation of x^2=4ay can be re-expressed in the parametric form:
x=2at
y=at^2
Converting parametric to cartesian:
1) set up a simultaneous equation
2) from (1) make t the subject
3) sub t into (2) and simplify
Converting cartesian to parametric
1) Express the cartesian equation in x^2 = 4ay to find a
2) sub a into each line of x=2at and y=at^2
Locus and the Parabola Rules
Rule 1
Coordinates
Any point, P, on the parabola x^2=4ay, has the coordinates, P(2ap, ap^2)
Locus and the Parabola Rules
Rule 2
Chord
If P(2ap, ap^2) and Q(2aq, aq^2) are any two points on the parabola x^2=4ay, then the chord PQ has: i) gradient: m= p+q/2 ii) equation: y-1/2(p+q)x+apq=0 Proof: i) gradient formula with P and Q ii) Point gradient formula
Locus and the Parabola Rules
Rule 3
Focal Chord
If PQ is a focal chord then pq=-1
Proof:
Sub focus (0,a) into the PQ equation
Locus and the Parabola Rules
Rule 4
Tangent
The tangent to the parabola x^2=4ay at the point P(2ap, ap^2) has i) gradient: m=p ii) equation: y-px+ap^2=0 Proof i) derivative ii) point gradient formula
Locus and the Parabola Rules
Rule 5
Tangent Intersection
The tangents to the parabola x^2=4ay at the points P(2ap, ap^2) and Q(2aq, aq^2) intersect at the point (a(p+q),apq)
Proof:
2 tangents simultaneous equation
Locus and the Parabola Rules
Rule 6
Normal
The normal to the parabola x^2=4ay at the point P(2ap, ap^2) has i) gradient: m= -1/p ii) equation: x+py= ap^3+2ap Proof i) m1m2=-1 to tangent gradient ii) point gradient formula
Locus and the Parabola Rules
Rule 7
Normal Intersection
The normals to the parabola x^2=4ay at the points P(2ap, ap^2) and Q(2aq, aq^2) intersect at the point
(-apq(p+q), a(p^2+pq+q^2+2))
Proof
Simultaneous equations of normals
Locus and the Parabola Rules
Rule 8
Cartesian Tangent
If point A(x1, y1) lies on the parabola x^2=4ay then the equation of the tangent at A is given by
x x1=2a(y+y1)
Proof
Gradient (derivative) then point gradient formulaLocus and the Parabola Rules
Rule 9
Cartesian Normal
If point A(x1,y1) lies on the parabola x^2=4ay then the normal at A is given by
y-y1=-2a/x1(x-x1)
Proof
Derivative, m1m2=-1 of tangent then point gradient formula
Locus and the Parabola Rules
Rule 10
Chord of Contact
The equation of the chord of contact, XY, of tangents drawn from the point P(x1, y1) to the parabola x^2=4ay is given by
x x1= 2a(y+y1)
Proof
Equation of chord XY
Intersection of tangents equal to P
x1=a(p+q)—1 y1=apq—2
Find (p+q), apq to simplify the equation of the chord
