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Find the value of p if the line joining (p, 4) and (6, -2) is…

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MATHS

Find the value of p if the line joining (p, 4) and (6, -2) is perpendicular to the line joining (2, 9) and (-1, 3)

A. 4

B. 3

C. 

D. 6

QUICK ANSWER…

D

DETAILS…

If two lines are perpendicular to each other, then it follows that the multiplication of their gradients will be equal to minus one.

That is,

if the gradient of one of the lines is = m,

and the gradient of the second line is = m

then, m₁ × m₂ = – 1

from the question, the gradient of the first line m₁ = (change in y)/(change in x)

m₁ = (y₂ – y₁)/(x₂ – x₁)

where

x₁ = p

y₁ = 4

x₂ = 6

y₂ = -2

m₁ = (-2 – 4)/(6 – p)

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m₁ = -6/(6 – p)

also for the second line, m₂ = (y₂ – y₁)/(x₂ – x₁)

where

x₁ = 2

y₁ = 9

x₂ = -1

y₂ = 3

m₂ = (3 – 9)/(-1 – 2)

m₂ = -6/-3 = 2

applying the rule: m₁m₂ = -1

-6/(6 – p) × 2 = -1

Doing the math to determine p, we have:

-12/(6 – p) = -1

-12 = -1(6 – p)

12 = 6 – p

Rearranging to take p the other side and 12 the other way,

p = 6 – 12

p = -6

Now for the right answer to the above question:

  1. Option A is incorrect.
  2. Option B is incorrect.
  3. C is not correct.
  4. D is the correct answer.

KEY-POINTS…

You may please note these/this:

  • If two lines are perpendicular, then the product of their gradients must be = -1
  • m₁m₂ = -1, where m = gradient of line 1, m = gradient of line 2.
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/ culled from 2020 JAMB-UTME mathematics past question 22 /

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