A straight line L has a y-intercept of -4 and is perpendicular to the line segment connect… — JAMB UTME

A straight line $L$ has a $y$ -intercept of $-4$ and is perpendicular to the line segment connecting the points $P(-3, 1)$ and $Q(5, 5)$ . If the line $L$ intersects the $x$ -axis at $(a, 0)$ , what is the value of $a$ ?

A
$-8$
$-2$ Answer
C
$2$
D
$8$

Answer

The value of

a

-2

The gradient of

PQ

is calculated as

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

. Because line

L

is perpendicular to

PQ

, its slope is

m_2 = -2

. Combining this with the given

y

-intercept of

-4

yields the equation

y = -2x - 4

. Setting

y = 0

to determine the

x

-intercept gives

0 = -2a - 4

, which solves to

a = -2

Step-by-Step Solution

Find the gradient

m_1

of the line segment

PQ

m_1 = \frac{5 - 1}{5 - (-3)} = \frac{4}{8} = \frac{1}{2}

The formula for the gradient between two points

(x_1, y_1)

and

(x_2, y_2)

m = \frac{y_2 - y_1}{x_2 - x_1}

Calculate the gradient

m_2

of line

L

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

Perpendicular lines have gradients satisfying

m_1 \cdot m_2 = -1

Determine the equation of line

L

y = -2x - 4

Using the slope-intercept form

y = mx + c

, where

m = -2

and

c = -4

Find the

x

-intercept by substituting

y = 0

and

x = a

0 = -2a - 4 \implies 2a = -4 \implies a = -2

At the

x

-axis, the

y

-coordinate is always equal to

0

Key Concept

Perpendicular Line Slopes and Coordinate Intercepts

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic