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# In the xy-plane, a circle C is drawn with center at (1, 2) and radius

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Re: In the xy-plane, a circle C is drawn with center at (1, 2) and radius [#permalink]
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ktzsikka wrote:
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 sufficient to answer the question asked.
b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Equation of the circle that can be derived from the prompt : $$(x-1)^2 + (y-2)^2 = 5^2$$

1) Point (a,b) lies on line L, so if we can figure out if this is a point on the circle as well, then we can confirm whether L is tangent to the circle or not.

Upon expanding the equation provided, we get $$a^2 + b^2 - 2a - 4b <= 20$$ ...(1)
Kinda stumped here, maybe expanding the circle's equation will reveal something.
On expanding the circle's equation we get $$x^2 + y^2 - 2x - 4y = 20$$ ...(2)

This looks similar to (1), with the x in place of a, and y in place of b. However, note the "<=20" in (1), there will be some values of (a,b) that make the LHS equal to 20, and some that make it lesser than 20, so we can't say with certainty that L is tangent to the circle. If it were just "=20", this would be the answer, as we would be able to say with certainty that a point that lies on L lies on the circle as well. Eliminate A and D.

2) X intercept of L is 10. Based on the y intercept, L could pass through the circle, be tangent to it, or not touch it at all. Not sufficient.

1 & 2
Don't see how these two statements can work together to give the answer. Statement 1 doesn't say anything about the y intercept and statement 2 has no bearing on 1 anyway.

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Re: In the xy-plane, a circle C is drawn with center at (1, 2) and radius [#permalink]
ShreyasJavahar, what do you mean by 'X intercept of L is 10. Based on the y intercept, L could pass through the circle, be tangent to it, or not touch it at all'. cant visualize if you can elaborate and explain
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Re: In the xy-plane, a circle C is drawn with center at (1, 2) and radius [#permalink]
Dhwanii wrote:
ShreyasJavahar, what do you mean by 'X intercept of L is 10. Based on the y intercept, L could pass through the circle, be tangent to it, or not touch it at all'. cant visualize if you can elaborate and explain

The circle in the attachment isn't the circle mentioned in the question, I just included an image from Google, but it'll work just fine as what I'm going to explain will hold true for the circle mentioned in the question as well.
Now, let's assume the X intercept of a certain line is 8 (it is 10 in the question). So we have a line that passes through 8 on the X axis, what about the Y intercept? If the Y intercept is -2, the line will pass through the circle; if the Y intercept is -100, the line will not even touch the circle; for some particular value of the Y intercept, the line will be tangent to the circle. The three aforementioned samples outline what I meant by the phrase you have quoted. Hope this helps.
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Re: In the xy-plane, a circle C is drawn with center at (1, 2) and radius [#permalink]
ShreyasJavahar correct me if I'm wrong in order to determine whether line will pass through the circle, be tangent to it or not touch circle at all we need information on both x intercept and y intercept. Because statement of our question has info on only one of the intercept (X INTERCEPT) and not the other it is insufficient. Thanks a a lot for your time and help
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Re: In the xy-plane, a circle C is drawn with center at (1, 2) and radius [#permalink]
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Dhwanii wrote:
ShreyasJavahar correct me if I'm wrong in order to determine whether line will pass through the circle, be tangent to it or not touch circle at all we need information on both x intercept and y intercept. Because statement of our question has info on only one of the intercept (X INTERCEPT) and not the other it is insufficient. Thanks a a lot for your time and help

That's right. The Y intercept could be anywhere, and that means the line could be anywhere in relation to the circle. Since we have multiple outcomes, the option is insufficient.
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Re: In the xy-plane, a circle C is drawn with center at (1, 2) and radius [#permalink]
@ktzsikkaGiven: In the xy-plane, a circle C is drawn with center at (1, 2) and radius equal to 5.
Asked: Is line l a tangent to the circle C?

Equation of the circle: -
(x-1)^2 + (y-2)^2 = 5^2 = 25
x^2 - 2x + 1 + y^2 - 4y + 4 = 25
x(x-2) + y(y-4) = 20

(1) Point A with coordinates (a, b) lies on line l such that a(a-2) +b(b-4) ≤ 20.
Point A lies on circle C if a(a-2) + b(b-4) = 20 and l may be a tangent to circle C otherwise it does not lie on circle C and l is not a tangent.
NOT SUFFICIENT

(2) The x-intercept of line l is 10.
Equation of line l: x/10 + y/k = 1: where k is y-intercept.
kx + 10y = 10k
Since k may vary and l is not unique
NOT SUFFICIENT

(1) + (2)
(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.
Equation of line l: x/10 + y/k = 1: where k is y-intercept.
kx + 10y = 10k
Since point(a,b) lies on line l
ka + 10b = 10k
k(10-a) = 10b
k = 10b/(10-a)
Equation of line l:
10bx/(10-a) + 10y = 10*10b/(10-a)
10bx + 10(10-a)y - 100b = 0 :  $$a\neq 10$$
Still it can not be ascertained whether line l is a tangent to the circle C or not.
NOT SUFFICIENT

IMO E
Re: In the xy-plane, a circle C is drawn with center at (1, 2) and radius [#permalink]
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