A fixed line segment AB has a length of 8\text{ cm}. A point P moves in the plane such tha… — JAMB UTME

A fixed line segment $AB$ has a length of $8\text{ cm}$ . A point $P$ moves in the plane such that the area of $\triangle PAB$ is always $20\text{ cm}^2$ . Which of the following best describes the locus of $P$ ?

A pair of parallel lines at a perpendicular distance of $5\text{ cm}$ on opposite sides of $AB$ Answer
B
A circle of radius $5\text{ cm}$ centered at the midpoint of $AB$
C
A pair of parallel lines at a perpendicular distance of $2.5\text{ cm}$ on opposite sides of $AB$
D
The perpendicular bisector of the line segment $AB$

Answer

A pair of parallel lines at a perpendicular distance of

5\text{ cm}

on opposite sides of

AB

The area of

\triangle PAB

is given by

\frac{1}{2} \times \text{base} \times \text{height}

. With a base

AB = 8\text{ cm}

, an area of

20\text{ cm}^2

requires a constant height

h = 5\text{ cm}

. The geometric locus of all points at a constant distance from a given straight line consists of two parallel lines situated at that distance on either side of the line.

Step-by-Step Solution

Express the area formula of the triangle in terms of base and height.

\text{Area} = \frac{1}{2} \times \text{base} \times h = \frac{1}{2} \times 8 \times h = 4h

The base of

\triangle PAB

is fixed as the length of segment

AB

, which is

8\text{ cm}

Calculate the constant perpendicular height

h

4h = 20 \implies h = 5\text{ cm}

Setting the calculated area equal to the given constant area of

20\text{ cm}^2

gives the required height.

Determine the geometric locus corresponding to a constant perpendicular height.

The locus of points at a fixed distance

h = 5\text{ cm}

from line

AB

is a pair of parallel lines running on either side of

AB

at a distance of

5\text{ cm}

Any point

P

lying on either of these two parallel lines maintains a perpendicular distance of

5\text{ cm}

from

AB

, ensuring

\text{Area}(\triangle PAB) = 20\text{ cm}^2

Key Concept

Locus at a constant distance from a straight line

Question

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Step-by-Step Solution

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