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

Circle Related Long Answer Type Questions (11-20) | S N Dey Mathematics
11. Show that the circle touches the co-ordinate axes. Also find the equation of the circle which passes through the common points of intersection of the above circle and the straight line
and which also passes through the origin.
Solution.

The equation of the given circle can be written as
From we get the centre of the circle
is
and radius
unit.
Now, if , we get from the given equation of the circle,
As we have just one point common between axis and circle, it means it touches
axis at
Similarly, if , we get from the given equation of the circle,
As we have just one point common between axis and circle, it means it touches
axis at
Hence, the given circle touches the two axes.
Now, the equation of the circle passing through the intersection of the given circle and the straight line is
Since the circle passes through the point
, we get from
Hence, the equation of the circle is
12. A circle through the common points of the circles and
has its centre on the line
Find the centre and radius of the circle.
Solution.
Any circle through the common points of the circles and
can be written as
Comparing with
, we get
So, centre of the circle is
Since the centre of the circle lies on the line
,
Hence, the centre of the circle
Also, the radius of the circle is
13. The circle and the line
intersect at
and
. Find the equation of the circle on
as diameter.
Solution.
The equation of the given straight line
The equation of the given circle is
From we get,
From and
we get,
Now, by we get for
Similarly, for
So, the equation of the circle on as diameter is given by
14. Find the equation to the circle described on the common chord of the circles and
as a diameter.
Solution.
Two given equations of circles are
So, by we get the equation of common chord which is
Equation of any circle through the intersection of two given circles are
From we get the centre of circle
which lies on the common chord
Now, putting the value of in
we get
15. Find the equation to the locus of the mid-points of chords drawn through the point on the circle
Solution.
Let be the chord passing through the point
Any point on the circle
can be written as
If is the co-ordinates of mid-point of
, then
Then from we get,
Hence, by we can say that the equation to the locus of the mid-points of chords is
16. A circle passes through the origin and intersects the co-ordinate axes at
and
. If the length of the diameter of the circle be
unit, then find the locus of the centroid of the triangle
Solution.
Any circle passing through the origin can be written as
Let
Since the circle passes through the points
so
If be the centroid of
, then
So, by we get,
the locus of the centroid of the triangle
is
17. Find the equation of a circle circumscribing the triangle whose sides are and
If
vary so that
, find the locus of the centre of the circle.
Solution.


18.Find the area of the equilateral triangle inscribed in the circle
Solution.

The equation of the given circle is
The centre of the circle is
and radius
Now, from the figure , we note that in
So, the area of the equilateral triangle inscribed in the circle is
19. Find the area of the equilateral triangle inscribed in the circle
Solution.
Comparing the given circle with the standard form of the circle we get,
So, the radius

Now, from the figure () we notice that
So, the length of the side of
is
So, the area of the equilateral triangle inscribed in the circle
20. Prove analytically that the straight line joining the middle point of a chord of a circle of a circle with the centre is perpendicular to the chord.
Solution.

To prove : Given that
From and
, we get
Hence,