# Circle | Part-7 | S N Dey

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

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