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I.E.S. Izpisa BelmonteSeccin Europea

Matemticas I Seccin Europea

65LOCUS.CONICS.

LOCUS(Lugar geomtrico).

A locus is a set of points that satisfy a certain condition or criteria.The criteria that defines the locus has to be translated to an algebraic

language in order to solve problems involving types of locus.

Perpendicular bisector: The perpendicular bisector of a line segmentis the locus of points on the plane that are equidistant from the endpoints.

Angle bisectors The bisector of an angle is the locus of points on the planethat are equidistant from the rays that form the angle.

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66CIRCLE.Definition.

The equation of a circle is the locus of points on the plane that are

equidistant from a fixed point called the centre .This common distance iscalled radiusEquation.

The equation of the circle whose centre is point O(a,b) and radius r is:

222 rbyax .

An equation of the type 022 CByAxyx can be a circle if

022

22

C

BA, then its centre is point

2,

2

BA and its radius is

CBA

22

22

.

Circle passing through three non-aligned points.Given three non-aligned points, there is a unique circle passing through

them. In order to determine its centre, we have to calculate the point ofintersection of the perpendicular bisector of the line segments determined forthe three points. The radius will be the distance from the centre to one ofthese points.
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67Circle-line intersection (Posiciones relativas de recta ycircunferencia).

To find the common points of a circle and a line, solve the systems

formed by the equations of both.In general, a quadratic equation is obtained, which will have a sign for the

discriminant, , depending on the following solutions:

Secant Line

> 0 .

Two solutions: There are two points

of intersection.

Tangent line

= 0.

One solution: the line is a tangent tothe circle.

< 0.

No solution: There is no intersectionbetween the line and circle.

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68ELIPSE.Definition.The ellipse is the locusof points on the plane whose sum of distances to two

fixed points, foci, are always constant.

Elements of the Ellipse

Foci

The foci are the fixed points of the ellipse which are located on the major axis. They are

denoted by F and F'.

Major Axis

The major axis of the ellipse is the line segment , which has a length of 2a.

Minor Axis

The minor axis of the ellipse is the line segment , which has a length of 2b.

Focal Length

The focal length of the ellipse is the line segment , which has a length of 2c.

Centre

The centre of the ellipse is the point of intersection of the axes. It is the centre of

symmetry of the ellipse.

Vertices

The vertices of the ellipse are the points of intersection of the ellipse with the axes. They

are denoted by A, A', B and B'.

Focal Radii

The focal radii are the line segments that join a point on the ellipse with both foci. They

are denoted by PF and PF'.

Semi-Major Axis

The semi-major axis is the line segment that runs from the centre of the ellipse, through a

focus, and to a vertex of the ellipse. Its length is a.
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69Semi-Minor Axis

The semi-minor axis is the line segment, perpendicular to the semi-major axis, which

runs from the centre of the ellipse to a vertex. Its length is b.

If a = b, an ellipse is more accurately defined as a circle.

Axes of Symmetry

The axes of symmetry are the lines that coincide with the major and minor axes.

Relationship between the Semiaxes

Centre Majoraxis

Graph Equation Foci

(0,0) HorizontalF'(-c, 0) andF(c, 0)

(0,0) VerticalF'(0, -c) andF(0, c)

(x0,y0) HorizontalF(x0-c,y0) andF(x0+c,y0)

(x0,y0) Vertical

F'(x0,y0c)

and F(x0,y+c)

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70The eccentricity(excentricidad) of an ellipse is a number that expresses thedegree of roundness of the ellipse.

HYPERBOLA.Definition.

The hyperbola is the locus of points on the plane whose difference ofdistances to two fixed points, foci, are constant.

|d(P,F)-d(P,F)|=2a

Elements of the HyperbolaFociThe foci are the fixed points of the hyperbola. They are denoted by F and F'.Transverse Axis or real axisThe transverse axis is the line segment between the foci. Conjugate Axisorimaginary axis

The conjugate axis is the perpendicular bisector of the line segment

(transverse axis).CentreThe centre is the point of intersection of the axes and is also the centre ofsymmetry of the hyperbola.

VerticesThe points A and A' are the points of intersection of the hyperbola with thetransverse axis.Focal RadiiThe focal radii are the line segments that join a point on the hyperbola withthe foci: PFand PF'.

Focal LengthThe focal length is the line segment , which has a length of 2c.
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71Semi-Major AxisThe semi-major axis is the line segment that runs from the centre to a vertexof the hyperbola. Its length is a.Semi-Minor Axis

The semi-minor axis is a line segment which is perpendicular to the semi-major axis. Its length is b.Axes of SymmetryThe axes of symmetry are the lines that coincide with the transversal andconjugate axis.AsymptotesThe asymptotes are the lines with the equations:

The Relationship between the Semiaxes:

Eccentricity measures the degree of the opening of the branches of thehyperbola.

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72Centre,Real

axisGraph Equation Foci

C(0,0)

Horizontal

F'(-c, 0) andF(c, 0)

C(0,0)

Vertical

F'(0, -c) andF(0, c)

(x0,y0)

Horizontal

F(x0-c,y0) andF(x0+c,y0)

(x0,y0)

Vertical

F'(x0,y0c) and

F(x0,y+c)

PARABOLA.Definition.

The parabola is the locus of points on the plane that is equidistant froma fixed point called the focus and a fixed line called the directrix.
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73Elements of the Parabola.

Focus

The focus is the fixed point F.Directrix

The directrix is the fixed line d.

Focal Parameter

The focal parameter is the distance from the focus to the directrix. It is

denoted by p.

Axis

The axis is the line perpendicular to the directrix that passes through the

focus.

Vertex

The vertex is the point of intersection of the parabola with its axis.

Vertex, focusand directrix

Axis Graph Equation

V(0,0)

F(p/2,0)

d:x=-p/2

x-axis

V(0,0)

F(-p/2,0)

d:x=p/2

x-axis

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