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12 Jan 2022, 04:30
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
55% (02:23) correct
45%
(02:33)
wrong
based on 60
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In the xy-plane, a circle C is drawn with center at (1, 2) and radius equal to 5. Is line l a tangent to the circle C?
(1) Point A with coordinates (a, b) lies on line l such that a(a-2)+ b(b-4) ≤ 20.
(2) The x-intercept of line l is 10.
(1) Point A with coordinates (a, b) lies on line l such that a(a-2)+ b(b-4) ≤ 20.
(2) The x-intercept of line l is 10.
kungfury42
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05 Feb 2022, 15:21
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Bunuel
I think this is a high-quality question 💯💯
I'll start here with statement 2 since it seems slightly less tricky and easier to evaluate
Quote:
This statement essentially tells us just that the line L passes through the point (10, 0) and nothing else. We know there are INFINITE number of lines that pass through any given point on the Cartesian plane. Could one of those lines possibly be a tangent to the circle? Possibly yes. But is every line passing through that point a tangent to the circle? Absolutely no. So we don't know whether this line L is a tangent to the circle.
Usually this would've sufficed to say that statement 2 is insufficient to answer our question, however, there is a small catch here. The catch is that we must first ensure that the point (10,0) lies on or outside the circle.
Because if it lies inside the circle, not a single line that passes through it, would under any chance, be a tangent to the given circle.
Distance of (10,0) from the center of the circle (1,2) = sqrt{(10-1)²+(2-0)²} = sqrt{9² + 4²} = sqrt (85) ~~ 9.2
Since this distance is greater than the radius of the circle we can comfortably say that the given point lies outside the circle and that the line L passing through it may or may not be a tangent to the circle. Clearly, insufficient statement, eliminate options B and D.
Quote:
Let's rearrange what's given here to form something more recognizable and friendly.
a(a-2) + b(b-4) ≤ 20
a²-2a + b²-4b ≤ 20
a²-2a+1 + b²-4b+4 ≤ 20+1+4 (completing the squares)
(a-1)² + (b-2)² ≤ 25
(a-1)² + (b-2)² ≤ 5²
Ahaa! This inequation now tells me that my point (a, b) is such that it's distance from the center of the circle (1, 2) is less than or equal to 5 units (the radius of the circle) which means that point (a, b) is such that it either lies on the boundary of the circle or inside the circle.
IF the point (a, b) were inside the circle there's no chance that any line L passing through it would be a tangent to the circle. However, IF the point (a, b) lies on the boundary of the circle, there might a one in an infinite chance that one particular line L passing through it would be tangent to our circle. Hence, by just saying that point (a, b) lies on or within the circle, we cannot say whether a line L passing through this point would be a tangent to the circle. Clearly insufficient, eliminate A.
Quote:
Quote:
Combining the two statements we know that the line L passes through (10, 0) and another point (a, b) which lies on or within the circle.
If the point lies within the circle, there's no chance that line L would be a tangent to the circle. However, even if the point did lie on the circle, we don't know whether the line L passing through (10, 0) and (a, b) will cut the circle at 1 unique point or 2 different points. We cannot answer this unless we know the specific values of a and b. Clearly, even both the statements together are insufficient. Eliminate C.
Hence, our answer is option E. This question cannot be answered even after using both the statements together.
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28 Apr 2023, 07:10
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