Question

If the points (p, 1), (1, 2) and (0, q + 1) are collinear, show that 1/p +1/q,=1​

Answer 1

Answer:

Consider the gradients of all the possible three lines

Step-by-step explanation:

take:

( p , 1 ) → A

( 1 , 2) → B

( 0 , q + 1 ) → C

Now the three possible points are AB, BC, and AC

note that since the points are collinear, they all should have the same gradient. Now, recall that the gradient of a line is give by:

→ { y₂ - y₁ / x₂ - x₁ } and denoted by "m"

now lets find the gradient of all the three possible lines, keeping in mind that all three gradients should be the same.

1. AB: m₁ = 2-1/1-p = 1/1-p

2. BC: m = (q+1)-2/(0-1) = q-1/-1

3. AC: m = (q+1)-1/(0-p) = q/-p

Now since we know that m = m = m

Let's first equate m = m ;

→  1 / ( 1 - p ) = ( q - 1 ) / ( -1 )

Cross multiplying :  → ( q - 1 ) ( 1 - p ) = - 1

                                  → q - pq - 1 + p = - 1

                                  → q + p - pq = 0

                                  → ( q + p ) / ( pq ) = 1

                                  → q/pq + p/pq = 1

And finally  { 1/p +  1/q = 1 }