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.
d)EACH 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.
Answer E.