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# The circle with center (1,0) and radius 5 intercepts at points A and B

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GMATH Teacher
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Joined: 12 Oct 2010
Posts: 935
The circle with center (1,0) and radius 5 intercepts at points A and B  [#permalink]

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06 Feb 2019, 07:40
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Difficulty:

55% (hard)

Question Stats:

58% (02:57) correct 42% (02:49) wrong based on 19 sessions

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GMATH practice exercise (Quant Class 20)

The circle with center (1,0) and radius 5 intercepts at points A and B the straight line defined by C = (0,0) and D= (4,8). If M = (w,z) is the midpoint of the line segment AB, what is the value of w+z ?

(A) 3/7
(B) 1/2
(C) 3/5
(D) 3/4
(E) 1

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The circle with center (1,0) and radius 5 intercepts at points A and B  [#permalink]

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06 Feb 2019, 08:46
1
fskilnik wrote:
GMATH practice exercise (Quant Class 20)

The circle with center (1,0) and radius 5 intercepts at points A and B the straight line defined by C = (0,0) and D= (4,8). If M = (w,z) is the midpoint of the line segment AB, what is the value of w+z ?

(A) 3/7
(B) 1/2
(C) 3/5
(D) 3/4
(E) 1

Eqn of circle $$(x-a)^2+(y-b)^2=r^2 => (x-1)^2+(y)^2=5^2$$
eqn of line y=mx since line is passing through (0,0) =>putting the values (4,8) we get $$y=2x$$
solving these equations we will get point A and B
(12/5,24/5) and (-2,-4)
midpoints will be for X $$(12/5+(-2))/2=1/5$$ and for Y $$(24/5-4)/2=2/5$$

points are(1/5,2/5) sum= 3/5
C
GMATH Teacher
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Re: The circle with center (1,0) and radius 5 intercepts at points A and B  [#permalink]

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07 Feb 2019, 06:43
fskilnik wrote:
GMATH practice exercise (Quant Class 20)

The circle with center (1,0) and radius 5 intercepts at points A and B the straight line defined by C = (0,0) and D= (4,8). If M = (w,z) is the midpoint of the line segment AB, what is the value of w+z ?

(A) 3/7
(B) 1/2
(C) 3/5
(D) 3/4
(E) 1

$$? = w + z = {x_M} + {y_M}\,\,\,\,\,\left[ {M = \left( {{x_M};{y_M}} \right)} \right]$$

$$A,B\,\,\,\,\left\{ \matrix{ \,{\left( {x - 1} \right)^2} + {y^2} = {5^2}\,\,\,\,\,\left[ {{\rm{circle}}} \right] \hfill \cr \,y = 2x\,\,\,\,\,\left[ {{\rm{line}}} \right] \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\left( {x - 1} \right)^2} + {\left( {2x} \right)^2} = {5^2}\,\,\,\, \Rightarrow \,\,\,\,5{x^2} - 2x - 24 = 0\,\,\,\,\left( * \right)$$

$$M = \left( {{{{x_A} + {x_B}} \over 2}\,\,;\,\,{{{y_A} + {y_B}} \over 2}} \right)\,\,\,\mathop = \limits^{M\, \in \,\,{\rm{line}}} \,\,\,\left( {{{{x_A} + {x_B}} \over 2}\,\,;\,\,{{2\,{x_A} + 2\,{x_B}} \over 2}} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = {3 \over 2}\left( {{x_A} + {x_B}} \right)$$

$$\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{x_A} + {x_B} = {\rm{sum}}\,\,{\rm{roots}} = - {{\left( { - 2} \right)} \over 5} = {2 \over 5}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = {3 \over 2}\left( {{2 \over 5}} \right) = {3 \over 5}$$

The correct answer is therefore (C).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: The circle with center (1,0) and radius 5 intercepts at points A and B   [#permalink] 07 Feb 2019, 06:43
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# The circle with center (1,0) and radius 5 intercepts at points A and B

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