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There are two circles with centers A and B having radii 5 cm and 3 cm.

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There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 18:04
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A
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C
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There are two circles with centers A and B having radii 5 cm and 3 cm. They touch each other internally. If the perpendicular bisector of segment AB meets the bigger circle at P and Q. Find the length of PQ.

A. 2√3
B. 3√4
C. 4√6
D. 5√6
E. 5√7

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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post Updated on: 07 Jun 2020, 03:47
If, as per figure, PDA is right angle triangle, then
PD^2 + DA^2 = AP^2
PD^2 =25-1=24
PD= 2root6.
PQ = 2*PD = 4root6

Imo. C
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Originally posted by Raxit85 on 06 Jun 2020, 18:20.
Last edited by Raxit85 on 07 Jun 2020, 03:47, edited 2 times in total.
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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 18:33
ANSWER:C
LET BOTH CIRCLE TOUCH INTERNALLY AT POINT T .
And let R be the point at which PQ bisect AB
AB=AT-BT-5-3=2
AR=2/1=1
Length of PQ=2PR=2X sqrt PA^2 - AR^2
PQ=2 X Sqrt 25-1=2 sqrt 24=4 sqrt 6
=2 X Sqrt
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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 19:01
Let O be the point on AB that bisects AB. AP and AQ would be the radius of the larger circle which is 5.
AB is the difference between the radius of the larger circle and the smaller circle which is 5-3= 2
Thus we have two right triangles with hypotenuse as 5 and one leg as 1.
Thus OP^2= OQ^2= 5^2- 1^2
OP= OQ= square root of 24
Thus, PQ= 2* square root of 24
PQ= 4√6

Answer: C
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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 20:18
given are two circles with centers A and B having radii 5 cm and 3 cm touching internally
The perpendicular bisector of segment AB meets the bigger circle at P and Q. Target Find the length of PQ.
Let the point of touch of two circles be C ; so AC = 5cm and BC = 3cm
also a LINE PQ which for circle center A pass through the line AC and intersect it at point D
we get AC-BC = AB ; 5-3 ; 2 cm
and PQ is perpendicular bisector so AD = 1 cm
now for ∆ PAD ; PA = 5cm and AD = 1 cm we can determine PD ; i.e 25-1 ; 24 ; 2√6
and since PQ is a chord of circle so its distance will be 2 *PD ; 2*2√6 ; 4√6
OPTION C


There are two circles with centers A and B having radii 5 cm and 3 cm. They touch each other internally. If the perpendicular bisector of segment AB meets the bigger circle at P and Q. Find the length of PQ.

A. 2√3
B. 3√4
C. 4√6
D. 5√6
E. 5√7
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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 20:59
There are two circles with centers A and B having radii 5 cm and 3 cm. They touch each other internally. If the perpendicular bisector of segment AB meets the bigger circle at P and Q. Find the length of PQ.

A. 2√3
B. 3√4
C. 4√6
D. 5√6
E. 5√7

Solution:

Attachment:
Picture2.png
Picture2.png [ 8.23 KiB | Viewed 313 times ]


OAP is a right-angled triangle, in which, OA = 1 and AP = 5,

Therefore, \(OP^2 = AP^2 - AO^2 = 25 - 1 ; OP = \sqrt{24} = 2\sqrt{6}\)
PQ = 2 OP = \(4\sqrt{6}\)

So, C is the correct answer
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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 21:15
Quote:
There are two circles with centers A and B having radii 5 cm and 3 cm. They touch each other internally. If the perpendicular bisector of segment AB meets the bigger circle at P and Q. Find the length of PQ.

A. 2√3
B. 3√4
C. 4√6
D. 5√6
E. 5√7


length of AB = 2cm

perpendicular bisector of AB divides AB in two equal parts at point O.
in one of triangles formed by perpendicular bisector,
AP^2 = AO^2 + OP^2
OP^2 = 25 - 1 = 24; OP = 2√6

similarly, in other triangle, OQ also = 2√6

PQ = OP + OQ = 4√6
Ans: C
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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 21:23
Let line segment AB and PQ cut each other at M
Let AM = X, BM = Y, PM = QM = Z
In triangle APM,
AP^2 = AM^2+PM^2
5^2 = X^2+Z^2
25-X^2 = Z^2 -------(1)

BP^2 = BM^2+PM^2
3^2 = Y^2+Z^2
9-Y^2 = Z^2 -------(2)

25-X^2 = 9-Y^2
16 = X^2-Y^2 --------(3)

A. 2√3 = 2Z => Z = √3
Putting value of Z in equation 1&2 respectively
25-3 = 22 = X^2 &
9-3 = 6 = Y^2.
Again putting value of X^2 & Y^2 in equation 3
22-6 = 16 => 16 = 16 (matching)

B. 3√4 = 2Z => Z = 3
Putting value of Z in equation 1&2 respectively
25-9 = 16 = X^2 &
9-9 = 0 = (zero value, which cannot be null)

C. 4√6 = 2Z => Z = 2√6
Putting value of Z in equation 1&2 respectively
25-12 = 13 = X^2 &
9-12 = (negative value, which cannot be negative)

D. 5√6 = 2Z => Z = 5√6/2
Putting value of Z in equation 1&2 respectively
25-75 = (negative value, which cannot be negative)

E. 5√7 = 2Z => Z = 5√7/2
Putting value of Z in equation 1&2 respectively
25-87.5 = (negative value, which cannot be negative)

IMO A

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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 21:56
OA is 4root6 . again pythagoras theorem is to be used
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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post Updated on: 06 Jun 2020, 23:41
Given:
1. There are two circles with centers A and B having radii 5 cm and 3 cm.
2. They touch each other internally.
3. The perpendicular bisector of segment AB meets the bigger circle at P and Q.

Asked: Find the length of PQ.

Attachment:
Screenshot 2020-06-07 at 1.06.04 PM.png
Screenshot 2020-06-07 at 1.06.04 PM.png [ 30.75 KiB | Viewed 265 times ]


AB = 5 - 3 = 2
\(AT = TB = \frac{AB}{2} = 1\)

\(In \triangle PTB\)

\(\angle PTB = 90^0\)
TB = 1
PB = radius of bigger circle = 5
\(PT = \sqrt{PB^2 - TB^2} = \sqrt{5^2 - 1^2} = \sqrt{24} = 2\sqrt{6}\)

\(PQ = PT*2 = 4\sqrt{6}\)

IMO C
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Originally posted by Kinshook on 06 Jun 2020, 23:40.
Last edited by Kinshook on 06 Jun 2020, 23:41, edited 1 time in total.
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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 06 Jun 2020, 23:41
Quote:
There are two circles with centers A and B having radii 5 cm and 3 cm. They touch each other internally. If the perpendicular bisector of segment AB meets the bigger circle at P and Q. Find the length of PQ.

A. 2√3
B. 3√4
C. 4√6
D. 5√6
E. 5√7

c,IMO

AB = 2 . So PQ/2 = root (24). so PQ = 2 root(24) = 2 * 2 (root (6)) = 4 root(6).
So c .

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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.  [#permalink]

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New post 07 Jun 2020, 00:35
Let the meeting point of two circles be Y.

AY=AB+BY..........(AY=Radii of the bigger circle 5 cm. BY=Radii of the smaller circle 3cm)
\(\therefore AB = 5-3 = 2cm.\)

A perpendicular bisector of segment AB will divide AB & itself into two equal parts...(AX=XB, PX=XQ) Let the perpendicular bisector meet AB at X.
\(\therefore AX=XB=1cm\)

In \\(triangle APX\), AP = Radii of bigger circle 5cm, AX=1cm.
By Pythagoras theorem, \(PX=2\sqrt 6\)

PQ = PX+XQ
PQ = 2PX

\(PQ= 4 \sqrt 6\)
Thus, PQ is (C) \(4 \sqrt6\)

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Re: There are two circles with centers A and B having radii 5 cm and 3 cm.   [#permalink] 07 Jun 2020, 00:35

There are two circles with centers A and B having radii 5 cm and 3 cm.

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