A straight line L_1 passes through the points A(-2, 1) and B(4, 5). A second line L_2 is p… — JAMB UTME

A straight line $L_1$ passes through the points $A(-2, 1)$ and $B(4, 5)$ . A second line $L_2$ is perpendicular to $L_1$ and passes through the midpoint of the line segment $AB$ . If $L_2$ intersects the $x$ -axis at point $R$ , and $R$ divides the line segment joining $P(1, -2)$ and $Q(k, 4)$ internally in the ratio $1 : 2$ , what is the value of $k$ ?

$7$ Answer
B
$4$
C
$-5$
D
$5$

Answer

The value of

k

7

The midpoint of

AB

(1, 3)

and the gradient of

L_1

\frac{2}{3}

. The gradient of the perpendicular line

L_2

-\frac{3}{2}

, yielding the equation

3x + 2y - 9 = 0

. Setting

y = 0

gives the

x

-intercept

R(3, 0)

. Applying the internal section formula for ratio

1 : 2

on the

x

-coordinates yields

\frac{k + 2}{3} = 3

, which solves directly to

k = 7

Step-by-Step Solution

Find the midpoint

M

of line segment

AB

and the gradient of line

L_1

Midpoint

M = \left(\frac{-2 + 4}{2}, \frac{1 + 5}{2}\right) = (1, 3)

. Gradient

m_1 = \frac{5 - 1}{4 - (-2)} = \frac{4}{6} = \frac{2}{3}

Line

L_2

passes through the midpoint

M

and its orientation depends on the gradient of

L_1

Determine the gradient

m_2

L_2

and its equation.

Since

L_2 \perp L_1

m_2 = -\frac{1}{m_1} = -\frac{3}{2}

. Equation of

L_2

y - 3 = -\frac{3}{2}(x - 1) \implies 3x + 2y - 9 = 0

Perpendicular lines have gradients whose product is

-1

(

m_1 \cdot m_2 = -1

Find the coordinates of point

R

, the

x

-intercept of

L_2

Set

y = 0

3x + 2y - 9 = 0 \implies 3x - 9 = 0 \implies x = 3

. Thus,

R = (3, 0)

The

x

-intercept occurs where

y = 0

on the coordinate plane.

Apply the section formula to find

k

Point

R(3, 0)

divides

P(1, -2)

and

Q(k, 4)

in ratio

1 : 2

. The

x

-coordinate is given by

x_R = \frac{1(k) + 2(1)}{1 + 2} = \frac{k + 2}{3}

. Setting

\frac{k + 2}{3} = 3

gives

k + 2 = 9 \implies k = 7

The internal section formula states that a point dividing

(x_1, y_1)

and

(x_2, y_2)

in ratio

m : n

has coordinates

\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)

Key Concept

Perpendicular lines, midpoints, intercepts, and section formula in coordinate geometry

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