In the previous article , we have completed the discussions about the Short Answer Type Questions. In this article, we will discuss 10 Long Answer type Questions from Chhaya Mathematics , Class 11 (S N De book ).

Long Answer Type Questions (1-10) | Circle | S N Dey Solution
1.Find the equation to the circles which touch the axis and pass through
and
Solution.

The equation of the circle touching the axis can be written as
where
is the centre of the circle with radius
unit.
Since the circle passes through ,
Again, since the circle passes through ,
Subtracting from
we get
Now, from and
we get,
, corresponding values of
So, the required equation of circle (for ) is
and also the required equation of circle (for ) is
2. If two straight lines and
lie along two diameters of a circle , which touches the
axis, find the equation of the circle.
Solution.
According to the problem, center of the circle will be the point where both lines cross each other. Now, solving two straight lines and
we get,

So, the centre of the circle
Since, the circle touches axis, so the equation of the circle is
3. A circle touches the lines and
If the centre of the circle lies in the first quadrant , show that there are two such circles and find their equations . Specify which of these is inscribed within the triangle formed by the given lines.
Solution.
Since the circle touches both axis and
axis , the centre of the circle is
So, the equation of the circle is
Since is a tangent to the circle
so the distance of
from the centre
of the circle = the radius of the circle.
So, the equation of the circle where
Now, since the points and
lie on the same side of the straight line
, the circle with radius
unit is inscribed within the triangle formed by the given lines.
4. Find the equations to the circles which which touch the axis of at a distance
from the origin and intercept a length
unit on the axis of
Solution.

Let the equation of the circle be
Now, the the length of the intercept on axis
By we get two circles by two corresponding values of
Using
two circles are given by
5. A circle passes through the point and touches the straight line
at the point
. Find its equation.
Solution.

6. Find the equation of the circle which touches the axis at a distance
unit from the origin and cuts off an intercept of length
unit from the
axis.
Solution.

Let the equation of the circle be
The circle can be written as
The length of the intercept on axis
So, by and
we get the equation of the circle as follows :
7. Show that the circles and
touch each other externally ; find the co-ordinates of their point of contact.
Solution.
Two given circles are
Comparing with
we get,
So, for the circle centre
and radius
Comparing with
we get,
So, for the circle centre
and radius
Distance () between the centres of two circles
So, two circles touch each other externally.
Again, so that the co-ordinates of their point of contact is
8. Prove that the circles and
touch each other internally. Find the equation of their common tangent.
Solution.
Two given circles are
Comparing with
we get,
So, for the circle centre
and radius
In a similar way, from we get, the centre
and radius
Distance () between the centres of two circles
Hence, by we can conclude that two circles touch each other internally.
Now, subtracting from
we get
So, the equation in , represents the equation of their common tangent.
9. If the circles and
touch each other , prove that ,
Solution.
Two given circles are
The centre of the circle is
and the radius
The centre of the circle is
and the radius
10. Prove that the circles and
touch each other. Find the co-ordinates of the point of contact.
Solution.
Two given equation of circles are and
Clearly, centre of circle is
and radius is

Again, the circle can be rewritten as
ans so centre of circle
is
Also, the radius () of circle of
is
Distance between the centres of two circles are
So, from and
we get,
So, two circles touch each other internally.
From the figure, we notice that
Suppose that two circles touch each other at the point
From we get
Hence, the co-ordinates of the point of contact is
To download full PDF solution of Circle (Chhaya Mathematics, Class 11 ), click here.