For the two circles ${{x}^{2}}+{{y}^{2}}=16$ and ${{x}^{2}}+{{y}^{2}}-2y=0$ , there is/are:

(a) One pair of common tangents

(b) Two pair of common tangents

(c) Three common tangents

(d) No common tangent

Hint: For solving this question first we will simplify the equation of the given circles and then find the coordinates of their centre and their radius. Then, we will plot them and try to find out how many common tangents will be there.

Complete step-by-step solution:

Given:

We have equation of following two circles:

1. Circle A: ${{x}^{2}}+{{y}^{2}}=16$ .

2. Circle B: ${{x}^{2}}+{{y}^{2}}-2y=0$ .

Now, we will simplify the equation of circle B. Then,

$ {{x}^{2}}+{{y}^{2}}-2y=0 $

$ \Rightarrow {{x}^{2}}+{{y}^{2}}+1-2y=1 $

${{x}^{2}}+{{\left( y-1 \right)}^{2}}=1$

Now, the equations of the circle B can be written as ${{x}^{2}}+{{\left( y-1 \right)}^{2}}=1$ .

Now, as we know that ${{\left( x-a \right)}^{2}}+{{\left( y-b \right)}^{2}}={{r}^{2}}$ is the equation of a circle whose centre is at point $\left( a,b \right)$ and radius of the circle is $r$ . Then, the radius and centre of the given two circles can be determined. There radius and centres are written below:

1. Circle A: ${{x}^{2}}+{{y}^{2}}=16$ , the radius will be $4$ units and the centre will be at the origin $\left( 0,0 \right)$ .

2. Circle B: ${{x}^{2}}+{{\left( y-1 \right)}^{2}}=1$ , the radius will be $1$ unit and centre will on the $y-axis$ at $\left( 0,1 \right)$ .

Now, from the above data, we can plot the circle A and circle B. Their plot is shown below:

Now, from the above plot, we conclude circle B is completely inside circle A and if you find the distance between their centres B (0,1) and A (0,0) then it will be 1 unit which is less than the sum of their radii. So, we can conclude that there won’t be any common tangent between circle A and circle B.

Thus, there will be no common tangent between the given equation of circles.

Hence, (d) is the correct option.

Note: Here, the student should proceed stepwise to find the correct answer easily. Firstly the coordinates of the centre of the given circles should be determined and then their radii. Moreover, there can be one more method in which we can answer without plotting them in which if the distance between the centres is less than the sum of the radius then no common tangent will be there.