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.

Related Topics

Teachers and parents can guide the student to go through the pages and get the good knowledge of parabola, ellipse and hyperbola. If you have any doubt, please contact us through mail, we will help you to clear your doubt.

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