Two statements are given below - one labelled Assertion (A) and the other labelled Reason(R). Read the statements carefully and choose the option that correctly describes statements (A)and (R).Assertion (A): The origin is the ONLY point equidistant from (2, 3) and (-2, -3).Reason (R): The origin is the midpoint of the line joining (2, 3) and (-2, -3).A) Both (A) and (R) are true and (R) is the correct explanation for (A).B) Both (A) and (R) are true but (R) is not the correct explanation for (A).C) (A) is true but (R) is false.D) (A) is false but (R) is true.
Question
Two statements are given below - one labelled Assertion (A) and the other labelled Reason(R). Read the statements carefully and choose the option that correctly describes statements (A)and (R).Assertion (A): The origin is the ONLY point equidistant from (2, 3) and (-2, -3).Reason (R): The origin is the midpoint of the line joining (2, 3) and (-2, -3).A) Both (A) and (R) are true and (R) is the correct explanation for (A).B) Both (A) and (R) are true but (R) is not the correct explanation for (A).C) (A) is true but (R) is false.D) (A) is false but (R) is true.
Solution
To answer this question, we first need to understand the concepts of midpoint and equidistant points.
The midpoint of two points (x1, y1) and (x2, y2) is given by ((x1+x2)/2, (y1+y2)/2).
So, the midpoint of the points (2, 3) and (-2, -3) is ((2-2)/2, (3-(-3))/2) = (0, 3). This means that the origin (0, 0) is not the midpoint of the line joining (2, 3) and (-2, -3). Therefore, statement (R) is false.
Now, let's consider the concept of equidistant points. A point is said to be equidistant from two other points if its distance from each of the points is the same.
The distance between two points (x1, y1) and (x2, y2) is given by √((x2-x1)² + (y2-y1)²).
So, the distance between the origin (0, 0) and the point (2, 3) is √((2-0)² + (3-0)²) = √(4 + 9) = √13. Similarly, the distance between the origin (0, 0) and the point (-2, -3) is √((0-(-2))² + (0-(-3))²) = √(4 + 9) = √13.
This means that the origin is indeed equidistant from the points (2, 3) and (-2, -3). Therefore, statement (A) is true.
So, the correct option is C) (A) is true but (R) is false.
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