Question

Which of the following circles have their centers on the x-axis? Check all that apply. A.(x - 0)2 + (y - 5)2 = 49 B.(x - 0)2 + (y - 0)2 = 25 C.(x - 4)2 + (y - 0)2 = 22 D.(x + 1)2 + (y - 7)2 = 16

I NEED HELP ASAPPP

Answer 1

Explanation:

Solution

verified

Verified by Toppr

The centres and radii of the circles are

C

1

​

(1,3) and r

1

​

=

1+9−9

​

=1.

C

2

​

(−3,1) and r

2

​

=

9+1−1

​

=3.

C

1

​

C

2

​

=

20

​

,r

1

​

+r

2

​

=4=

16

​

∴C

1

​

C

2

​

>r

1

​

+r

2

​

. Hence the circles are non-intersecting. Thus there will be four common tangents.

Transverse common tangents are tangents drawn from the point P which divides C

1

​

C

2

​

internally in the ratio of radii 1:3.

Co-ordinates of P are

(

1+3

1(−3)+3.1

​

,

1+3

1.1+3.3

​

) i.e. (0,

2

5

​

).

Direct common tangents are tangents drawn from the point Q which divides C

1

​

C

2

​

externally in the ratio 1:3.

Co-ordinates of Q are tangents through the point P(0,5/2).

Any line through (0,5/2) is

y−5/2=mx.....(1)

or mx−y+5/2=0.

Apply the usual condition of tangency to any of the circle

∴

(m

2

+1)

​

m.1−3+5/2

​

=1

or (m−

2

1

​

)

2

=m

2

+1

or −m−3/4=0 or 0m

2

−m−3/4=0.

Hence m=−3/4 and ∞ as coeff. of m

2

is zero.

Therefore from (1),

x

y−5/2

​

=m=∞ and −3/4.

∴x=0 is a tangent and y−5/2=−3x/4

or 3x+4y−10=0 is another tangent.

Direct tangents are tangents drawn from the point Q(3,4).

Now proceeding as for transverse tangents their equations are

y=4,4x−3y=0.