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In the coordinate system, the center of a circle lies at (2, 3). If point A with coordinates (-1, 7) does not lie outside the circle, which of the following points must lie inside the circle?
I. (0, 7)
II. (5, -1)
III. (-2, 7)
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I and II only
  • e)
    None of the above
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
In the coordinate system, the center of a circle lies at (2, 3). If po...
Given
  • Circle with center at (2, 3)
    • Let’s assume the radius of the circle to be r.
  • Point A(-1, 7) does not lie outside the circle
To Find: Which of the points in the options must lie inside the circle?
 
Approach
  1. To know the points that must line inside the circle, we need to find the distance between the center of the circle and the point (x, y)
 
  • Let us understand how can we find the distance between 2 points in the coordinate plance.
  • Let us say the above figure represent the coordinates of one point → point A → (a,b) and of another point → Point C → (x,y) in the co-ordinate system
    1. Distance of point A from point B along the x-axis = (x-a)
      1. So, AB=x-a
    2. Distance of point C from point B along the y-axis = (y-b)
      1. So, CB= y-b
    3. Now, the triangle ABC formed is a right angled triangle, so by Pythagoras theorem
In general, it can be said that the distance between any two points is =
 So we can find the distance between Point A and the centre of the circle by the above formula
  • If this distance between the center of the circle and point(x, y) is less than the minimum possible value of r, then the point (x, y) must lie inside the circle.
    • So, we need to find the minimum possible value of r.
  • We are given that point A does not lie outside the circle. So, it may lie either inside the circle or on the circle.
    1. Hence (the distance of point A from the center) ≤ r
        1. That is, r ≥ (the distance of point A from the center)
      1. So, (minimum possible value of r) = (the distance of point A from the center)
        1. We will use the above relation to find the minimum possible value of r.
Working Out
  1. Coordinates of point A = ( -1, 7) and coordinates of center of the circle = (2, 3)
    1. Distance of point A from the center of the circle ​
    2. Thus the minimum possible value of r = 5
  2. Evaluating the 3 options
    1. Distance of point (0, 7) from the center (2, 3)
      1. ​​​
  3. As the distance of point is less than the minimum possible value of r, this point must lie inside the circle
    1. Distance of point (5, -1) from the center (2, 3)
      1. As the distance of point is not less than the minimum possible value of r, this point may or may not lie inside the circle.
        1. For example, if r > 5, the point will lie inside the circle
        2. If r = 5, the point will not lie inside the circle, it will lie on the circle.
      2. Distance of point (-2, 7) from the center (2, 3)
        1. As the distance of point is not less than the minimum possible value of r, this point may or may not lie inside the circle
Hence, we see that only option I (0, 7) always lie inside the circle.
Answer: A
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Most Upvoted Answer
In the coordinate system, the center of a circle lies at (2, 3). If po...
To determine which of the given points must lie inside the circle, we need to calculate the distance between each point and the center of the circle. If the distance is less than the radius of the circle, then the point lies inside the circle.

1. Calculating the distance for point (0, 7):
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, the distance between (0, 7) and (2, 3) is:
distance = sqrt((2 - 0)^2 + (3 - 7)^2) = sqrt(4 + 16) = sqrt(20) = 2sqrt(5)

Since 2sqrt(5) is greater than the radius of the circle, point (0, 7) lies outside the circle.

2. Calculating the distance for point (5, -1):
The distance between (5, -1) and (2, 3) is:
distance = sqrt((2 - 5)^2 + (3 - (-1))^2) = sqrt(9 + 16) = sqrt(25) = 5

Since 5 is equal to the radius of the circle, point (5, -1) lies on the circumference of the circle, but not inside it.

3. Calculating the distance for point (-2, 7):
The distance between (-2, 7) and (2, 3) is:
distance = sqrt((2 - (-2))^2 + (3 - 7)^2) = sqrt(16 + 16) = sqrt(32) = 4sqrt(2)

Since 4sqrt(2) is greater than the radius of the circle, point (-2, 7) lies outside the circle.

Therefore, out of the given points, only point A (-1, 7) lies inside the circle. Hence, the correct answer is option A (I only).
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