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
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
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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