These are the loci of points moving in a plane such that the ratio of it’s distances
from a fixed point and a fixed line always remains constant .
The ratio is called eccentricity(E).For Hyperbola, it is E>1 .

Problem 1 : A sample of gas is expanded in a cylinder
from 10 unit pressure to 1 unit pressure.Expansion follows
law PV=Constant.If initial volume being 1 unit, draw the
curve of expansion. Also Name the curve.

Construction

Step 1. Take pressure on vertical axis and Volume on horizontal axis.

Step 2. Divide both the axes in ten equal parts. Name these parts from 1 to 10.

Step 3. Now according to the table given below locate the points on the graph.

PROBLEM 2 : Point F is 50 mm from a line AB. A point P is moving in a plane such
that the ratio of it's distances from AB and line F remains constant and equal to
2/3. Draw locus of point P. { Eccentricity = 2/3 }

Construction

Step 1. Draw a vertical line AB and point F
50 mm from it. Divide 50 mm distance in 5 parts.

Step 2. Name 2nd part from AB as V. It is 20mm
and 30mm from AB and F line resp.
It is first point giving ratio of it’s
distances from AB and F 2/3 i.e 20/30.

Step 3. Form more points giving same ratio such
as 30/45, 40/60, 50/75 etc.
Taking 30,40 and 50 mm distances from
line AB, draw three vertical lines to the
right side of it.

Step 4. Now with 45, 60 and 75 mm distances in
compass cut these lines above and below,
with F as center.

Step 5. Join these points through V in smooth
curve.
This is required locus of P.It is the required hyperbola.