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

Question

GMATH  practice exercise (Quant Class 20)

The circle with center (1,0) and radius 5 intercepts
eqnof rhe circle = (x-1)^2+y^2 = 25 ......1
eqn of the line straight line defined by C = (0,0) and D= (4,8)
y=2x......
solving from 1 and 2 provides
(2/5,24/5) and (-2,-4)
solving for midpoint we get
(1/5,2/5)
=> sum = 3/5 hence IMO C

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

kunalcvrce · Answer

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

fskilnik · Answer

? = w + z = {x_M} + {y_M}\,\,\,\,\,\left

A,B\,\,\,\,\left\{ \matrix{
 \,{\left( {x - 1} \right)^2} + {y^2} = {5^2}\,\,\,\,\,\left  \hfill \cr 
 \,y = 2x\,\,\,\,\,\left  \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.

VENKYMECH · Answer

Assume that There is a circle and cord is formed with the points intercecting the circle with the line A and B.

Hence perpendicular from circle always intersects at the mid point. Just gind the perpendicular point from (1,0) on the line segment formed by origin and (4,8).

answer is 3/5

Posted from my mobile device

Crytiocanalyst · Answer

The circle with center (1,0) and radius 5 intercepts 
eqnof rhe circle = (x-1)^2+y^2 = 25 ......1
eqn of the line straight line defined by C = (0,0) and D= (4,8)
y=2x......
solving from 1 and 2 provides 
(2/5,24/5) and (-2,-4)
solving for midpoint we get 
(1/5,2/5)
=> sum = 3/5 hence IMO C

bumpbot · Answer

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

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