let C be a circle of radius 5 meters having centre at O.

Question

let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O. Then, the length of PQ, in meters, is nearest to

A 6.6
B 7.2
C 8.8
D 7.8
E 9

A 6.6
B 7.2
C 8.8
D 7.8
E 9

Kinshook · Answer

Given: Let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O.

Asked: Then, the length of PQ, in meters, is nearest to

Attachment:

Screenshot 2021-09-03 at 5.52.16 AM.png [ 44.76 KiB | Viewed 2066 times ]

Equation of the circle C:-
\(x^2 + y^2 = 5^2 = 25\)

Equation of line PQ passing through points A(0,4) & B(3,0): -
(y - 4) = (0-4)/(3-0) (x - 0)
y - 4 = -4x/3
4x + 3y = 12

Coordinate P & Q lies on the circle and on the line PQ so they must satisfy equation of the circle C and the line PQ.
\(x^2 + y^2 = 25\)
4x + 3y = 12
y = 4 - 4x/3

\(x^2 + (4 - 4x/3)^2 = 25\)
\(x^2 + 16 + 16x^2/9 - 32x/3 = 25\)
\(9x^2 + 144 + 16x^2 - 96x = 225\)
\(25x^2 - 96x - 81 = 0\)
\(x = \frac{(96 + - \sqrt{96^2 + 4*81*25})}{2*25} = \frac{(96 +- \sqrt{17316})}{50} = \frac{(96 + - 131.59)}{50} = (-.71, 4.55) \)

Point P : -
x = -.71
y = 4- 4x/3 = 4 + 4*.71/3 = 4.95
P = (-.71, 4.95)

Point Q: -
x = 4.55
y = 4 - 4x/3 = 4 - 4*4.55/3 = -2
Q = (4.55, -2)

Length of line segment PQ joining points P(-.71, 4.95) & Q(4.55, -2) : -
\(= \sqrt{(4.55+.71)^2 + (-2-4.95)^2} = \sqrt{75.97} = 8.8\)

IMO C

rohantewari · Answer

chetan2u   Bunuel

please help...

how can one do this more efficiently...cant know the values of the roots here

Crytiocanalyst · Answer

eqn of the line pq in 2 point form :-
(y - 4) / (x - 0) = (0-4)/(3-0)
y - 4 = -4 *x/3
4x + 3y = 12

pq satisfies the in the eqn of circle since the chords touch the circle 
x^2+y^2=25......1
y = 4 - 4x/3.........2

Therefore 2 in 1

x^2+(4−4x/3)^2=25
x2+16+16*x^2/9−32*x/3=25

25x^2−96x−81=0
x=-0,7 and y= 5 
corresponding values of X and y = 5 and y=-2

Which is approximately 9 therefore IMO C

chetan2u · Answer

rohantewari , 
A geometrical way, if it helps you.

I. Draw the figure as given.

II. Draw a perpendicular from O on chord PQ. 
It will bisect PQ. (property of chords)

III. Find the perpendicular OS. 
Various ways, but I would use AREA property to find OS.
Area of OAB =  \frac{1}{2}*OA*OB=\frac{1}{2}*AB*OS 
 \frac{1}{2}*4*3=\frac{1}{2}*5*OS........OS=2.4

IV. Find PS 
Take  	riangle OPS , which is a right angled triangle with sides OP=radius=5, and OS=2.4
PS= \sqrt{5^2-2.4^2} 
For ease of calculation, let me take 2.4~2.5= \frac{5}{2} 
PS= \sqrt{5^2-\frac{5}{2}^2}=\frac{5}{2}\sqrt{3}=2.5*1.73=4.325 
So, PS would be slightly more than 4.325, and PQ=2*4.325=8.7
 
If we take it as 2.4, then we will get answer as 8.8

C

let C be a circle of radius 5 meters having centre at O.

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let C be a circle of radius 5 meters having centre at O.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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