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

Question

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.

rockydx007 · Answer

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).

ktzsikka · Answer

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

ShreyasJavahar · Answer

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.

Answer E.

Dhwanii · Answer

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

ShreyasJavahar · Answer

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.

Dhwanii · Answer

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

ShreyasJavahar · Answer

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.

Kinshook · Answer

@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
eq 10 
Still it can not be ascertained whether line l is a tangent to the circle C or not.
NOT SUFFICIENT

IMO E

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

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GMAT Data Sufficiency (DS) Questions

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

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