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.Correct answer is option 'E'. Can you explain this answer?

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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 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.

Correct answer is option 'E'. Can you explain this answer?

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Test: Coordinate Geometry- 2

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

Steps 1 & 2: Understand Question and Draw Inferences

Circle C with center at (1, 2) and r = 5

To Find: Is line l a tangent to the circle?

For a line l to be a tangent to the circle, the circle C and the line l should share exactly one common point, i.e. a point lying on circle C should also lie on the line l
- For a point to lie on the circle C, the distance of the point from the center (1,2) should be exactly equal to radius of the circle i.e. 5 units.
Assuming the coordinates of a point lying on the circle to be (x, y), we can write

Using the same logic, if a point lies inside the circle, its distance from the center (1,2) should be less than 5 units.
Assuming the coordinates of a point lying inside the circle to be(x,y), we can write
Similarly, if a point lies outside the circle, its distance from the center (1,2) should be greater than 5 units.
Assuming the coordinates of a point lying outside the circle to be(x,y), we can write

- So, we can summarize the following:
  - A point (x, y) will lie inside the circle if x(x - 2)+ y(y - 4) = 20
  - A point (x, y) will lie on the circle if x(x - 2)+ y(y - 4) < 20
  - A point (x, y) will lie outside the circle if x(x - 2)+ y(y - 4) > 20

Step 3: Analyze Statement 1 independently

1.Statement 1 states that: "Point A with coordinates (a, b) lies on line l such that a(a-2) +b(b-4) ≤ 20".

As a(a-2) +b(b-4) ≤ 20, two cases may arise:
- If a(a-2) +b(b-4) = 20, the point A would lie on the circle. However, we cannot say if the line L meets circle C at only point A or at two different points.(one of which is Point A)
- If a(a-2) +b(b-4) < 20, point A would lie inside the circle. So, line l passes through the circle and hence is not a tangent to the circle C.
- Thus, Statement 1 is Insufficient to answer.

Step 4: Analyze Statement 2 independently

2. Statement 2 states that "The x-intercept of line l is 10".

The line l intersects the x- axis at point (10, 0)
Refer the below diagram:
- A line with x- intercept 10 may or may not be a tangent to the circle with center at (1,2)

Thus, Statement 2 is also Insufficient to answer

Step 5: Analyze Both Statements Together (if needed)

a(a-2) +b(b-4) ≤ 20
x-intercept of line l is 10.
Combination, of the two statements does not provide us with any extra information to answer the question.
Hence, both the statement together are also Insufficient to answer.

Thus, the correct answer is Option E .

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Most Upvoted Answer

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

Given information:
- Circle C has center at (1, 2) and radius 5.
- Line l is a line in the xy-plane.

To determine:
- Whether line l is a tangent to circle C.

Approach:
- To determine if line l is tangent to circle C, we need to find the point of intersection between the line and the circle.
- If there is only one point of intersection, then the line is a tangent. If there are two points of intersection, then the line is not a tangent.

Statement 1:
- Point A with coordinates (a, b) lies on line l such that a(a-2) + b(b-4) = 20.

Statement 2:
- The x-intercept of line l is 10.

Analysis of statement 1:
- The equation a(a-2) + b(b-4) = 20 represents a curve in the xy-plane.
- This curve could be a circle, an ellipse, or some other shape.
- Without any information about the line l, we cannot determine if it intersects the circle C at a single point or two points.
- Therefore, statement 1 alone is not sufficient to answer the question.

Analysis of statement 2:
- The x-intercept of a line is the x-coordinate of the point where the line intersects the x-axis.
- If the x-intercept of line l is 10, it means that the line passes through the point (10, 0).
- However, this information alone does not provide any insight into whether the line is tangent to the circle or intersects it at multiple points.
- Therefore, statement 2 alone is not sufficient to answer the question.

Conclusion:
- From the analysis of both statements, neither statement alone is sufficient to answer the question.
- Therefore, the correct answer is option E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data specific to the problem are needed.

Answered by Talent Skill Learning · View profile

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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 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.Correct answer is option 'E'. Can you explain this answer?

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