# Circle | Part-5 | S N Dey

In the previous article , we discussed few Short Answer Type Questions. In this article, we will discuss 9 more Short Answer type Questions from Chhaya Mathematics , Class 11 (S N De book ).

##### Short Answer Type Questions of Circle, S N Dey Mathematics, Class 11

21. A circle touches axis at and whose centre lies on the line find the equation of the circle.

Solution.

Let the radius of the circle be unit. The circle touches the axis at the point Now, since where and the centre of the circle is

Since the centre of the circle lies on

so

the centre of the circle is and the radius is unit.

the equation of the circle is

22. Find the equation of a circle which passes through the point and touches both the co-ordinate axes. How many of such circles are possible ?

Solution.

Let the radius of the circle be unit and the circle touches axis and axis at and respectively.

the co-ordinates of is

So, the equation of the circle can be written as

which passes through

Finally, by and , the equation of circle can be written as

or,

So, from the aforementioned discussions we can conclude that two such circles are possible.

23. Two circles of radii and units respectively pass through and touch both the and axes. Find the equations of the two circles . Also find the other common point of intersection.

Solution.

Since the circles passes through the point which lies in the first quadrant and since the given circles touch both the and axes, so the circles lie in the first quadrant.

Suppose that the circle with radius unit touches axis at the point and touches axis at .

Now from the following figure, we can notice that the co-ordinates of is

So, the equation of the circle

Similarly, we can determine that the equation of the circle with radius unit is given by

Now, we need to determine the other common point intersection.

Subtracting from we get,

From and we get that for the corresponding values of

So, common points of intersection of two circles are given by and

Hence, the other common point of intersection is

24. Prove that the lies on the circle Find the co-ordinates of the other extremity of the diameter through

Solution.

Putting in the equation of the circle we get and so the point lies on the given circle.

Comparing with the general form of the circle we get,

So, the centre of the given circle

Let the other extremity of the circle be

Since the the point lies on the circle, so the centre of the circle will be the midpoint of the straight line joining the points and which is

Hence, the co-ordinates of the other extremity of the diameter through is

25. Show that for all values of , the circle passes the point If varies , find the locus of the centre of the above circle.

Solution.

Since the given circle passes the point putting in we get

So, satisfy the given equation of the circle. So, the circle passes through the point for any values of

Comparing with the general form of the circle we get,

So, the centre of the circle

If is the centre of the circle, then

and

From we get

Finally, by we can conclude that the locus of the centre of the circle is

26. Find the co-ordinates of the points equidistant from the axes and lying on the circle

Solution.

The given equation of circle can be rewritten as :

Let the co-ordinates of the points equidistant from the axes be

Since the point lies on the given circle,

The co-ordinates of the points equidistant from the axes are

Again, since the point lies on the given circle,

So, by we get the imaginary values of and so we can discard those values.

27. Find the equation to the common chord of the two circles and and also find its length.

Solution.

The equation to the common chord of the two circles and is

From we get,

By and we get by replacing the value of

Let the roots of the equation in be

and

Let the chord intersects the circles at the points and .

and

So, the length of the chord is

28. Find the equation of the common chord of the two circles and Show that this chord is perpendicular to the line joining the centres of two circles.

Solution.

The equation of two given circles are and

Now, subtracting from we get

So, equation in represents the equation of the common chord of two circles.

The co-ordinates of centre of circle is and the co-ordinates of centre of circle is

Now, the slope of the straight line joining two centres of the circles is

Again , the slope of the straight line is

So,

Finally, by we can conclude that this chord is perpendicular to the line joining the centres of two circles.

29. Find the equation of the circle which passes through the origin and the points of intersection of the circles and

Solution.

The equation of the circle through the points of intersection of the circles and is

Since the circle passes through the origin, putting in we get,

the equation of the circle is