The perpendicular bisector of the line segment joining the points P(2, -1) and Q(6, 7) int… — JAMB UTME

The perpendicular bisector of the line segment joining the points $P(2, -1)$ and $Q(6, 7)$ intersects the $y$ -axis at $(0, c)$ . Find the value of $c$ .

Answer: 5

Answer

The value of

c

is 5.

The midpoint of

PQ

(4, 3)

and the gradient of

PQ

2

. The perpendicular bisector has a gradient of

-\frac{1}{2}

and passes through

(4, 3)

. Substituting these into the line equation gives

y - 3 = -\frac{1}{2}(x - 4)

, which simplifies to

y = -\frac{1}{2}x + 5

. The line intersects the

y

-axis at

(0, 5)

, so

c = 5

Step-by-Step Solution

Find the midpoint of the line segment

PQ

Midpoint

M = (4, 3)

The perpendicular bisector must pass through the midpoint of the segment.

Calculate the gradient of

PQ

Gradient

m_{PQ} = 2

Using the slope formula

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

for points

(2, -1)

and

(6, 7)

Determine the gradient of the perpendicular line

Perpendicular gradient

m_{\perp} = -\frac{1}{2}

Perpendicular lines have gradients that are negative reciprocals (

m_1 m_2 = -1

Formulate the equation of the perpendicular bisector and solve for the

y

-intercept

y = -\frac{1}{2}x + 5

, hence

c = 5

Using point-slope form

y - y_1 = m(x - x_1)

with point

(4, 3)

and

m = -\frac{1}{2}

, setting

x = 0

gives the

y

-intercept.

Key Concept

Perpendicular Bisector and Line Equations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic