Of all the points that satisfy the equation x2+ y2 =25approximately what percentage of the points also satisfy theinequality xy 0?a)25%b)33%c)50%d)75%e)80%Correct answer is option 'C'. Can you explain this answer?

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Of all the points that satisfy the equation x²+ y² =25 approximately what percentage of the points also satisfy the
inequality xy ≥ 0?

a)
25%
b)
33%
c)
50%
d)
75%
e)
80%

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

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Of all the points that satisfy the equation x2+ y2 =25approximately wh...

Given

x²+ y² =25

To Find: Percentage of points lying on the circle that satisfies xy ≥ 0?

Approach and Working

x² + y² = 25
- If we compare it with the distance formula:
  - ■ D² = [(xi - ^x2⁾² + (yi ^- y2)²]
  - ■ If (x₂, y₂) is considered as the origin, (x, y) as the other point
  - ■ We can write: (x-0)² + (y-0)² = 5²
  - ■ Thus, we are looking for all the points which are at a distance of 5 units from the origin.
- o If we try to plot it on a graph, we will notice that
  - This equation represents all the points lying on a Circle C with center(0,0) and radius of 5 units.
xy > 0, when the x and the y coordinates have the same signs, except for the case when either of x or y = 0
- x and y coordinates have the same signs in the 1^st and the 3^rdquadrant. So, we need to find the percentage of points of circle C that lie in the 1^st and the 3^rd quadrant.
A circle with center at origin will pass through all the 4 quadrants.
- So, if we find the arc length of the circle in the 1^st quadrant and the 3^rd quadrant, we can divide it by the total arc length of the circle (i.e. the circumference) to find the percentage of points satisfying the inequality xy > 0

We can see from the figure above that by symmetry, arc length of the circle in each quadrant = 1/4th of its total circumference = ¼ * 2πr
So, the total arc length present in the 1st and the 3rd quadrant = (¼ + ¼) *2πr = ½ *2πr
Hence, 50% of the points that lie on the circle, satisfy the inequality xy ≥0
And the correct answer is OPTION C

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Of all the points that satisfy the equation x2+ y2 =25approximately what percentage of the points also satisfy theinequality xy ≥ 0?a)25%b)33%c)50%d)75%e)80%Correct answer is option 'C'. Can you explain this answer?

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