Question 1: Find the locus of a point equidistant from the point and the y-axis.
Answer:
Let be the point which is equidistant from point
and y-axis.
Distant of point from y-axis is
.
Hence the locus of is
Question 2: Find the equation of the locus of a point which moves such that the ratio of its distances from and
is
.
Answer:
Given . Let
be the point which moves such that the ratio of its distances from
and
is
Given
Hence the locus of is
Question 3: A point moves as so that the difference of its distances from and
is
, prove that the equation to its locus
, where
.
Answer:
Given and
is
. Let
be the point.
Squaring both sides we get
Squaring once again we get
Hence locus of is
where
Question 4: Find the locus of a point such that the sum of its distances from and
is
.
Answer:
Given and
. Let
be the point such that the sum of its distances from
and
is
.
Squaring both sides
Squaring once again we get
Hence the locus of is
Question 5: Find the locus of a point which is equidistant from and x-axis.
Answer:
Given and x-axis. Let
be the point.
Hence the locus of is
Question 6: Find the locus of a point which moves such that its distance from the origin is three times its distance from x-axis.
Answer:
Let be the point. The distance of
from x-axis is
Hence the locus of is
Question 7: are two fixed points; find the equation to the locus of a point
which moves so that the area of the
is
units.
Answer:
Given . Let
be the point.
Area of
Therefore or
or
Hence the locus of or
Question 8: Find the locus of a point such that the line segments having end points and
subtend a right angle at that point.
Answer:
Given and
. Let
be the point such that
Hence the locus of is
Question 9: If and
are two fixed points, find the locus of a point
so that the area of
sq. units.
Answer:
Given and
. Let
be the point.
Therefore are of
or
or
Hence the locus of is
or
Question 10: A rod of length slides between the two perpendicular lines. Find the locus of the point on the rod which divides it in the ratio
.
Answer:
Let and
axis be the two perpendicular lines.
Let and
be the two intercepts on the axes.
Let be the point that divides
in
ratio.
and
Now
Hence locus of is
Question 11: Find the locus bf the mid-point of the portion of the line which is intercepted between the axes.
Answer:
Given line is
When
When
Let be th emid point of
We know
Hence the locus of is
Question 12: If is the origin and
is a variable point on
. Find the locus of the mid-point of
.
Answer:
Let be
. It lies on
… … … … … i)
Let be the mid point of
and
and
Substituting in i) we get
Hence the locus of is