Conic Sections

Conics or conic sections are the curves corresponding to various plane sections of a right circular cone by cutting that cone in different ways.

Each point lying on these curves satisfies a special condition, which actually leads us towards the mathematical definition ofconic sections.

If a point moves in plane in such a way that the ratio of its distance from a fixed point to its perpendicular distance from a fixed straight line, always remains constant, then the locus of that point I called a Conic Section.

The fixed point is called the focus and the fixed line is called directrix of the conic. The constant ratio is called theeccentricityand is denoted by e.

According to the value of there are three types o conic i.e. for e = 1, e < 1 and e > 1 the corresponding conic is calledparabola, ellipse and hyperbola respectively.

Aconic sectionor conic is the locus of a point, which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line, not passing through the fixed point.

The fixed point is called the focus.

• The fixed straight line is called thedirectrix.

• The constant ratio is called theeccentricityand is denoted by e.

• When theeccentricityis unity i.e. e = 1, the conic is called aparabola; when e < 1, the conic is called an ellipse; and when e > 1, the conic is called a hyperbola.

• The straight line passing through the focus and perpendicular to thedirectrixis called the axis of theparabola.

• The point of intersection of a conic with its axis is called vertex.

• The chord passing through focus and perpendicular to axis is calledlatus rectum.

• Any chord of theparabolawhich is perpendicular to the axis is called double ordinate.

• The straight line perpendicular to axis of theparabolapassing through vertex is called tangent at the vertex.

Axis of the conic:

The line through focus and perpendicular to thedirectrixis called the axis of the conic. The intersection point o conic with axis is known as the vertex of the conic.Enquiry: How do we mathematically define a parabola and what are its various features?

The locus of the point, which moves such that its distance from a fixed point (i.e. focus) is always equal to its distance from a fixed straight line (i.e. directrix), is calledparabola.

Equation of Parabola:

Let S be the focus, V be the vertex, ZM be thedirectrixand x-axis be the axis ofparabola. We require therefore the locus of a point P, which moves so that its distance from S, is always equal to PM i.e. its perpendicular distance from ZM. After appropriate configuration let S = (a, 0)

We have ten SP^{2}= PM^{2}

⇒ (x – a)^{2}+ y^{2}= (a + x)^{2}

⇒ y^{2}= 4ax This is the standardequation of a parabola.