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04 Sep 2024, 05:48
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
64% (01:58) correct
36%
(01:50)
wrong
based on 73
sessions
History
Date
Time
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Not Attempted Yet
The line \(x + y = 1\) intersects the circle \((x-2)^2 + (y+1)^2 = 8\) at which two points?
A. (1,0) and (-3,4)
B. (0,1) and (-3,4)
C. (0,1) and (4,-3)
D. (1,0) and (4,-3)
E. (2,-1) and (-1,2)
A. (1,0) and (-3,4)
B. (0,1) and (-3,4)
C. (0,1) and (4,-3)
D. (1,0) and (4,-3)
E. (2,-1) and (-1,2)
04 Sep 2024, 07:06
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I solved this in 1mins 24s by just plugging in the values. Plug (1,0) in the equation of the circle and it gives LHS != RHS => eliminate A and D.
Plug (0,1) in the equation of the circle, LHS = RHS. Also by just looking at (0,1) and equation of the line, we can say it will satisfy the equation. So now we only need to check (-3,4) and (4,-3). Both satisfy the equation of the line, so let's put these coordinates in the equation of the circle.
(-3,4) => (-5)^2 => eliminate this, since it will definitely give a value much greater than 8. So now C. (0,1) and (4,-3) is the only set of points that satisfy both the equation of the circle and the equation of line. But let's also check E now just to be safe.
(2,-1) satisfies the equation of line but not the equation of the circle so we can eliminate E.
Therefore, correct answer is (C).
Alternatively, you can solve this question by directly finding the points of intersection without looking at the options.
We have the line equation x + y = 1 and the circle equation (x-2)^2 + (y+1)^2 = 8.
To find the points of intersection, we need to solve the system of equations formed by the line and circle.
Substituting y = 1 - x into the circle equation, we get:
(x-2)^2 + (1-x+1)^2 = 8
(x-2)^2 + (2-x)^2 = 8
Simplifying this equation, we get:
x^2 - 4x + 4 + 4 - 4x + x^2 = 8
2x^2 - 8x = 0
Solving this quadratic equation, we get:
x = 0 or x = 4
Substituting these x-values back into the line equation, we get:
For x = 0, y = 1
For x = 4, y = -3
Therefore, the two points of intersection are (0,1) and (4,-3).
The correct answer is C.
I'd prefer the first approach in this particular question because of the easy values and fairly straight forward equations, but the second approach is more foolproof.
Plug (0,1) in the equation of the circle, LHS = RHS. Also by just looking at (0,1) and equation of the line, we can say it will satisfy the equation. So now we only need to check (-3,4) and (4,-3). Both satisfy the equation of the line, so let's put these coordinates in the equation of the circle.
(-3,4) => (-5)^2 => eliminate this, since it will definitely give a value much greater than 8. So now C. (0,1) and (4,-3) is the only set of points that satisfy both the equation of the circle and the equation of line. But let's also check E now just to be safe.
(2,-1) satisfies the equation of line but not the equation of the circle so we can eliminate E.
Therefore, correct answer is (C).
Alternatively, you can solve this question by directly finding the points of intersection without looking at the options.
We have the line equation x + y = 1 and the circle equation (x-2)^2 + (y+1)^2 = 8.
To find the points of intersection, we need to solve the system of equations formed by the line and circle.
Substituting y = 1 - x into the circle equation, we get:
(x-2)^2 + (1-x+1)^2 = 8
(x-2)^2 + (2-x)^2 = 8
Simplifying this equation, we get:
x^2 - 4x + 4 + 4 - 4x + x^2 = 8
2x^2 - 8x = 0
Solving this quadratic equation, we get:
x = 0 or x = 4
Substituting these x-values back into the line equation, we get:
For x = 0, y = 1
For x = 4, y = -3
Therefore, the two points of intersection are (0,1) and (4,-3).
The correct answer is C.
I'd prefer the first approach in this particular question because of the easy values and fairly straight forward equations, but the second approach is more foolproof.
04 Sep 2024, 08:14
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Thanks. I solved it already too, but it looks like i did something wrong when i tried to substitute in the equation of the circle instead of the first equation.
Let's start from this point:
X = 0
Or X = 4
And substitue in the equation of the circle
If X = 0, then Y = 1 or -3
If X = 4, then again Y = 1 or -3
Why did we take the pairs (0,1) and (4,-3) instead of (0,-3) and (4,1)? Although i solved it at the beginning then i got confused when i tried to substitute in the equation of the circle (regardless that the second pair is not in the answers, i am just trying to figure out whether the second pair is correct)
Edited:
Ohh, looks like i forgot it has to satisfy the two equations and not only the second one.
Posted from my mobile device
Let's start from this point:
X = 0
Or X = 4
And substitue in the equation of the circle
If X = 0, then Y = 1 or -3
If X = 4, then again Y = 1 or -3
Why did we take the pairs (0,1) and (4,-3) instead of (0,-3) and (4,1)? Although i solved it at the beginning then i got confused when i tried to substitute in the equation of the circle (regardless that the second pair is not in the answers, i am just trying to figure out whether the second pair is correct)
Edited:
Ohh, looks like i forgot it has to satisfy the two equations and not only the second one.
Posted from my mobile device
