Find Coordinate Geometry II formulae and important terms for quick revision. This unit includes chapters: Parabola, Ellipse and Hyperbola from Mathematics. This quick revision note is very important during examinations. It will help you in quick revision of complete unit in single go and in very less time. In some engineering entrance examinations such as UPSEE and WBJEE most of questions are asked directly on formula basis. This revision notes will help you in increasing your marks in this examination.
- A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix of the parabola and the fixed point F is called the focus.
- A line through the focus and perpendicular to the directrix is called the axis of the parabola.
- The point of intersection of parabola with the axis is called the vertex of the parabola.
- The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis or y-axis.
The four possible such orientations of parabola are shown below:
- Parabola is symmetric with respect to the axis of the parabola. If the equation has a y2 term, then the axis of symmetry is along the x-axis and if the equation has an x2 term, then the axis of symmetry is along the y-axis.
- When the axis of symmetry is along the x-axis the parabola opens to the
(a) right if the coefficient of x is positive,
(b) left if the coefficient of x is negative
- When the axis of symmetry is along the y-axis the parabola opens
(a) upwards if the coefficient of y is positive
(b) downwards if the coefficient of y is negative
- Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola.
- The length of latus rectum for the standard equation of parabola is 4a, where a is the coordinate of the focus.
- An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
P1F1 + P1F2 = P2F1 + P2F2 = P3F1 + P3F2
- The two fixed points in the definition of ellipse are called the foci.
- The midpoint of the line segment joining the foci is called the centre of the ellipse.
- The line segment through the foci of the ellipse is called the major axis.
- The line segment through the centre and perpendicular to the major axis is called the minor axis.
- The end points of the major axis are called the vertices of the ellipse.
- A circle described on major axis as diameter is called the auxiliary circle.
- The labeled diagram of the ellipse is shown as below
- Length of the major axis is denoted by 2a.
- Length of the minor axis is denoted by 2b.
- Distance between the foci is denoted by 2c.
- The equation of a hyperbola is simplest if the centre of the hyperbola is at the origin and the foci are on the x-axis or y-axis.
- The standard equations of hyperbola are:
- A hyperbola in which a = b is called an equilateral hyperbola.
- The standard equations of hyperbolas have transverse and conjugate axes as the coordinate axes and the centre at the origin.
- The foci are always on the transverse axis. It is the positive term whose denominator gives the transverse axis.
- Hyperbola is symmetric with respect to both the axes, since if (x, y) is a point on the hyperbola, then (– x, y), (x, – y) and (– x, – y) are also points on the hyperbola.
- Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.