In a coordinate plane, what is the total number of points 5 unita from the origin and equidistant from both the x and y axes?
1)1 2)2 3)3 4)4
two
the points (5/√2,5/√2) and (-5/√2,-5/√2)
To find the total number of points, 5 units from the origin and equidistant from both the x and y axes, we can use simple geometry.
The distance from the origin to any point on the coordinate plane can be found using the distance formula: d = sqrt((x2 − x1)² + (y2 − y1)²), where (x1, y1) is the coordinates of the origin and (x2, y2) is the coordinates of any other point.
Since we want to find points that are 5 units from the origin, we can set d = 5 in the distance formula:
5 = sqrt((x2 − 0)² + (y2 − 0)²)
25 = (x2² + y2²)
Now let's consider the condition that the points are equidistant from both the x and y axes. For a point to be equidistant from the x and y axes, the distances from the point to the x and y axes must be equal.
The distance from a point to the x axis is simply the absolute value of the y-coordinate, and the distance from a point to the y axis is the absolute value of the x-coordinate. So, for a point to be equidistant from both axes, we have the conditions:
|y2| = |x2|
Combining this condition with the equation we derived earlier:
25 = (x2² + y2²)
|y2| = |x2|
We can start by considering different possible values of x2 and solving for y2, considering the positive and negative values of x2:
Let's take x2 = 0:
From the equation 25 = (x2² + y2²), we have 25 = (0 + y2²). This implies y2² = 25, so y2 could be +5 or -5.
Now let's take x2 = 5:
From the equation 25 = (x2² + y2²), we have 25 = (5² + y2²). This implies y2² = 0, so y2 could only be 0.
Similarly, for x2 = -5, we get y2 = 0.
Therefore, there are a total of 3 points that are 5 units from the origin and equidistant from both the x and y axes. The points are (0, 5), (0, -5), and (5, 0).
So, the correct answer is 3), which corresponds to 3 points.