Question

The locus of the point \(\mathrm{P}\) which is at a constant distance of 2 units from the origin and which lies in the first or the second quadrants is (1) \(\mathrm{y}=-\sqrt{4-\mathrm{x}^{2}}\) (2) \(\mathrm{y}=\sqrt{4-\mathrm{x}^{2}}\) (3) \(\mathrm{x}=\sqrt{4-\mathrm{y}^{2}}\) (4) \(\mathrm{x}=-\sqrt{4-\mathrm{y}^{2}}\)The locus of the point \(\mathrm{P}\) which is at a constant distance of 2 units from the origin and which lies in the first or the second quadrants is (1) \(\mathrm{y}=-\sqrt{4-\mathrm{x}^{2}}\) (2) \(\mathrm{y}=\sqrt{4-\mathrm{x}^{2}}\) (3) \(\mathrm{x}=\sqrt{4-\mathrm{y}^{2}}\) (4) \(\mathrm{x}=-\sqrt{4-\mathrm{y}^{2}}\)

Short answer

Answer: The correct equation for the locus of point P at a constant distance of 2 units from the origin is y = √(4 - x²).

Step-by-step solution

Option 1

Option 1 states that the locus of point P is y = -√(4 - x²). However, since point P lies in either the first or the second quadrant, it has positive y-values. Thus, this option does not fit the given condition.

Option 2

Option 2 suggests that the locus of point P is y = √(4 - x²). For any given x value belonging to the first or second quadrant, the y-value will be positive. Let's use the distance formula to verify this option. The coordinates of point P are (x, y) and of the origin are (0, 0). Using the distance formula: \(2 = \sqrt{(x - 0)^2 + (y - 0)^2}\) Now, we know that y = √(4 - x²), let's substitute this into the equation: \(2 = \sqrt{x^2 + (4 - x^2)}\) Solving this, we get: \(4 = x^2 + 4 - x^2\) Which is true. Hence, option 2 is correct.

Option 3

The third option is x = √(4 - y²). Since point P lies in the first or the second quadrant, it should have non-negative x-values. We don't need to check this option further, since we've already found the correct option (Option 2).

Option 4

The fourth option is x = -√(4 - y²). Points P in the first and second quadrant will have non-negative x-values, so this option will not fit the condition. In conclusion, the correct equation for the locus of point P at a constant distance of 2 units from the origin is: \(\boxed{\mathrm{y} = \sqrt{4 - \mathrm{x}^2}}\) (Option 2)

Key concepts

Coordinate Geometry

In the realm of mathematics, coordinate geometry, also known as analytic geometry, is a potent tool that provides a connection between algebra and geometry through the use of an ordered pair of numbers known as coordinates. These coordinates are usually expressed as \( (x, y) \), where \( x \) represents the horizontal position, and \( y \) represents the vertical position on a two-dimensional grid.

Imagine that you're in an open field with a grid overlaid on it. If you were asked to find a particular spot, you could use the grid coordinates to pinpoint your exact location. This is the essence of coordinate geometry. The system allows us to solve geometric problems using algebraic equations, making it easier to understand the relationships between geometric figures.

For instance, you might be interested in finding all the points, or 'locus', that are a certain distance from a fixed point, like in our example with point \( \mathrm{P} \). Coordinate geometry equips us with formulas and methods to describe this collection of points through equations such as circle equations, line equations, and more.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four sections, known as quadrants, which are created by the intersection of the horizontal axis (x-axis) and the vertical axis (y-axis). Each quadrant represents a unique combination of positive and negative coordinates.

Quadrant I

Both x and y are positive. This is the top-right section of the plane.

Quadrant II

Here, x is negative, but y is positive, representing the top-left section.

Quadrant III

In this bottom-left section, both x and y are negative.

Quadrant IV

x is positive, and y is negative in the bottom-right section of the plane.

Understanding the quadrants is crucial when interpreting the location of a point, like our point \( \mathrm{P} \) which is situated in either the first or second quadrant. This implies that the y-value of point \( \mathrm{P} \) will always be positive, as it either lies in the top-right or top-left sections of the coordinate plane where the y-values are above the x-axis.

Circle Equation

The equation of a circle on a coordinate plane is a special form that represents all the points that are at a constant distance (the radius) from a fixed point (the center). In standard form, the circle equation is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center of the circle, and \( r \) is the radius.

For a circle centered at the origin \( (0, 0) \) with radius \( r \) the equation simplifies to: \[ x^2 + y^2 = r^2 \] When you're given a locus problem involving a point that remains a constant distance from the origin (just like our point \( \mathrm{P} \)), this circle equation becomes instrumental. By substituting \( 2 \) for the radius, the equation applicable to our example is \( y = \sqrt{4 - x^2} \), which outlines a semicircle in the first and second quadrants. The circle equation is at the heart of such locus problems and provides a straightforward way to mathematically visualize these points in space.