## Conics

Definition :

Conics is the locus of a point which moves in a plane so that its distance from a fixed point in the plane bears a constant ration to its distance from a fixed straight line in the plane. If S is a fixed point, l is a fixed straight line, P is a variable point moving in a way such that SP/PM is constant. Here the path traced by the variable point P is a conic. This constant is known eccentricity and it is denoted by e.  The fixed point S is called the focus and the fixed line l is called the directrix.      If e = 1, then the curve is a parabola.      If e < 1, then the curve is an ellipse.      If e > 1, then the curve is a hyperbola.

To prove that the equations of a conic is of degree 2 in x and y.

Let the focus be S(x₁,y₁) and the directrix be the line ax+by+c=0.

Let the eccentricity is e and P (x,y) be any point on it. Simplifying we get an equation of the form

ax²+2hxy+by²+2gx+2fy+c=0.

Therefore the equation is of second degree in x and y.

Now let us see how to find the type of the curve from the given equation of second degree. In the general equation Ax²+By²+Cx+Dy+F=0, with two variables x and y, then

• If both the variables are squared and multiplied by same number then the curve is a circle.
• If one of the variable is squared then it is a parabola.
• If squared variables have the opposite sign, then it is hyperbola.
• If both the variables are squared, have the same sign, but not multiplied by the same number then it is ellipse.

The new terms we learned here in this topic is

d - directrix  - the fixed line.

e - eccentricity-the constant distance.

f -  focus  -the fixed point from which the distance is measured.

We will discuss about the parabola, ellipse and hyperbola in the following pages.

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