Question

A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?

A. 0
B. 1
C. 2
D. Infinite

Answer 1

The third quadrant is the quadrant in which both the x and y coordinates are negative. Since the point is at a distance of 5 units from the origin, we can use the distance formula to find the set of points that satisfy this condition.

The distance formula is given by:
\[ \text{distance} = \sqrt{(x - 0)^2 + (y - 0)^2} \]

In this case, the distance is 5 units, so we have:
\[ 5 = \sqrt{x^2 + y^2} \]

Squaring both sides gives:
\[ 25 = x^2 + y^2 \]

The set of points that satisfy this equation form a circle centered at the origin with a radius of 5 units.

Now, for the points to lie in the third quadrant, both x and y must be negative. Therefore, the only point that satisfies these conditions is (-3, -4). So, there is 1 such point in the third quadrant.

Therefore, the answer is:
B

Answer 2

B. 1. There is 1 point that lies in the third quadrant.

Step-by-step explanation:

In the third quadrant, both x and y coordinates are negative. A point at a distance of 5 units from the origin can lie in the third quadrant only if both x and y coordinates are negative. Let's consider an example: (-3, -4). The distance between this point and the origin is calculated using the distance formula:

Distance = √((-3)^2 + (-4)^2) = √(9 + 16) = √25 = 5 units.

So, there is 1 point that lies in the third quadrant.