What is the equation of the locus of a point P(x, y) that moves in a plane such that its d… — JAMB UTME

What is the equation of the locus of a point $P(x, y)$ that moves in a plane such that its distance from the origin $(0,0)$ is always 5 units?

Cevap: x^2 + y^2 = 25 / x^2+y^2=25 / x²+y²=25

x^2 + y^2 = 25

By definition, the locus of a point moving at a fixed distance of 5 units from the origin

(0,0)

is a circle centered at

(0,0)

with radius 5. Substituting into the standard circle equation

x^2 + y^2 = r^2

yields

x^2 + y^2 = 5^2 = 25

Apply the distance formula between a general point

P(x, y)

and the origin

(0,0)

d = \sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2}

The distance between two points

(x_1, y_1)

and

(x_2, y_2)

\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Equate the distance formula to the given fixed distance of 5 units and square both sides.

\sqrt{x^2 + y^2} = 5 \implies x^2 + y^2 = 25

Squaring both sides eliminates the square root to give the algebraic equation of the locus.

The locus of points at a constant distance

r

from a fixed point

(h, k)

forms a circle with equation

(x-h)^2 + (y-k)^2 = r^2

Tahmini Süre:45s

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