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13 Nov 2021, 08:11
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Difficulty:
95%
(hard)
Question Stats:
43% (02:27) correct
57%
(02:25)
wrong
based on 197
sessions
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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.
(A) Statement 1 alone is sufficient, but statement 2 alone is not.
(B) Statement 2 alone is sufficient, but statement 1 alone is not.
(C) Both statements together are sufficient, but neither alone is sufficient.
(D) Each statement alone is sufficient.
(E) Neither statement is sufficient, even when combined.
(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.
(A) Statement 1 alone is sufficient, but statement 2 alone is not.
(B) Statement 2 alone is sufficient, but statement 1 alone is not.
(C) Both statements together are sufficient, but neither alone is sufficient.
(D) Each statement alone is sufficient.
(E) Neither statement is sufficient, even when combined.
14 Nov 2021, 11:17
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To find whether l is tangent or not, we need to prove that l is perpendicular to a line between center and the point where line l touches the circumference of the circle (if at all it does).
For a fact we know that two perpendicular lines have the product of their slope as -1.
Now, to know the slopes we need to know the equations of the line l and the line joining the center and the point where line l touches the circumference of the circle.
One more important thing to remember here is that we can find the equation of the circle using the circle coordinates and the radius. In this case (x-1)^2 + (y-2)^2 = 5^2, although you will find out that we do not need this information on this question.
Now that we are clear what we need to find, lets look at our resources.
1) Point A with coordinates (a, b) lies on line l such that a(a-2) +b(b-4) ≤ 20.
This does not tell us anything about a point where line l meets the circle.
If we dont know whether the line l touches the circle or not, we can not prove anything.
Hence Insufficient.
2)The x-intercept of line l is 10.
Same situation as A. No idea whther the line touches the circle or not.
1 & 2 together -> we still do not have any information about the line l touching the circle or not.
Hence both tother are insufficient, Answer is (E).
For a fact we know that two perpendicular lines have the product of their slope as -1.
Now, to know the slopes we need to know the equations of the line l and the line joining the center and the point where line l touches the circumference of the circle.
One more important thing to remember here is that we can find the equation of the circle using the circle coordinates and the radius. In this case (x-1)^2 + (y-2)^2 = 5^2, although you will find out that we do not need this information on this question.
Now that we are clear what we need to find, lets look at our resources.
1) Point A with coordinates (a, b) lies on line l such that a(a-2) +b(b-4) ≤ 20.
This does not tell us anything about a point where line l meets the circle.
If we dont know whether the line l touches the circle or not, we can not prove anything.
Hence Insufficient.
2)The x-intercept of line l is 10.
Same situation as A. No idea whther the line touches the circle or not.
1 & 2 together -> we still do not have any information about the line l touching the circle or not.
Hence both tother are insufficient, Answer is (E).
14 Nov 2021, 11:51
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rockydx007
Hi
Although your conclusion is correct but your treatment of statement 1(and for 1&2combined) is not thorough!
If you just open the brackets in the equation of circle (that you correctly mentioned), you will notice a pattern. Try it.
Will post the OE soon!
Best regards
Posted from my mobile device
