speech transcript

Liceo Scientifico Isaac Newton - Roma

the circle

in accordo con il

Ministero dell’Istruzione, Università, Ricerca

e sulla base delle

Politiche Linguistiche della Commissione Europea

percorso formativo a carattere

tematico-linguistico-didattico-metodologico

scuola secondaria di secondo grado

professor

Serenella Iacino

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The Circle

In our lesson today we will find the equation of the circle starting from its definition as a geometric locus.

We define a circle as the geometric locus of the points P(x,y) equidistant from a point C, which is the center , as shown in the picture and this is the general equation of a circle in two variables x and y and three coefficients a,b,c.

Starting from the general equation we must now study the different position of a circle on a Cartesian Plane, according to the variation of its coefficients a, b, c,:

  1. If a = b = 0, this is the general equation (x² + y² + c = 0), which representsa circle with the centre at the origin.
  2. If b = 0, the general equation becomes this one (x² + y² + ax + c = 0) and it represents a circle with the centre on the x axis.
  3. If a = 0, the general equation becomes this one (x² + y² + by + c = 0) which represents a circle with the centre on the y axis.
  4. If c = 0, this is the general equation (x² + y² + ax + by = 0) and it represents a circle passing through the origin.
  5. If a = c = 0 the general equation becomes this one (x² + y² + by = 0) and it represents a circle with the centre on the y axis and the circumference passing through the origin.
  6. And finally, if b = c = 0, the general equation becomes this one (x² + y² + ax = 0), which represents a circle with the centre on the x axis and the circumference passing through the origin.

To find the equation of a circle we need three independent conditions to determine the values of the constants, that is one condition for each constant. For example we can have the following conditions:

  1. The coordinates of the centre.
  2. The radius.
  3. The coordinates of a point P which the circle passes through: in this case the general equation of a circle (x² + y² + ax + by + c = 0) must be satisfied by putting the coordinates of the point P (xp; yp) in x and in y of the equation of the circle.

We can consider now the position of a straight line in relation to a circle. It could be:

  • Secant: if the distance from the centre is less than the radius.
  • Tangent: if the distance from the centre is equal to the radius.
  • External: if the distance from the centre is greater than the radius.

From an algebraic point of view, if we want to study the position of a straight line in relation to a circle, we must solve a system of equations: one for the circle, one for the straight line as we can see…….

From this system, we obtain an equation of the second degree, in which its discriminant may be:

  • Delta greater than zero : Δ > 0
  • Delta equal to zero: Δ = 0
  • Delta less than zero : Δ < 0

Now, we want to determine the equations of the two tangents, to a circle, from a given point P (x,y); and these two tangents will be:

  • Real: if the point (P) is outside the circle.
  • Coincident: if the point (P) is on the circumference.
  • Imaginary: if the point (P) is within the circle.

From an algebraic point of view, we must solve this system, composed by the equations of the circle and of the straight line passing through the point P.

Let’s consider now, the position of two circles relative to each other depending on the distance between their centres that can be:

  • External: if the distance (between their centres) is greater than the sum of the radii;
  • Tangent from the outside: if the distance (between their centres) is equal to the sum of the radii;
  • Secant: if the distance (between their centres) is less than the sum of the radii;
  • Tangent within: if the distance (between their centres) is equal to the difference between the radii;
  • Inside: if the distance (between their centres) is less than the difference between the radii;

This is the system composed by the equations of the two circles, from which we obtain the equation of the radical axis with the elimination method, as you can see from these illustrations.

Starting from the equations of two circles, first circle, second circle, we can obtain through a linear combination the equation of a set of circles.

From which, substituting ξ and ξ’ , we obtain this relation.

Grouping the terms of second degree, those of first degree and the costants, and dividing by 1+λ, we obtain this equation, which for different values of λ, represents all the circles passing through the intersections of ξ and ξ’ , which are the base points of the set, except for λ =-1, in fact, for this value λ = -1, we obtain the straight line….

This Radical Axis can be considered as a particular circle having an infinite radius, and can be called Degenerate Circle.

Materiale sviluppato da eniscuola nell’ambito del protocollo d’intesa con il MIUR

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