Question

if A,B,C are collinear points A=(3,4),B=(7,7)and AC=10 then C=

Answer 1

C can be (11,10) or (-5,-2)

Given, A is (3,4) and B is (7,7)

Let C be (x,y)

The distance between A and B is √[(7-3)²+ (7-4)²]

= 5

AB = 5 units

Given, AC is 10 units.

Slope of AB = (7-4)/(7-3)               [y₂-y₁/x₂-x₁]

= 3/4

So, slope of AC is (y-4)/(x-3)

This is equal to 3/4.

So, (y-4)/(x-3) = 3/4

⇒ 4y - 3x = 7

AC² = 10² = (y-4)² + (x-3)²

Putting (y-4) = 3(x-3)/4, we get,

So, AC² = [{3(x-3)/4}² + (x-3)²] = 10²

⇒ (x-3)²*(25/16) = 100

⇒ (x-3) = √(100*25/16) = ±8

For, +8,

x = 8+3 = 11, then y is (7+3x)/4 = (7+33)/4 = 10

Point is (11,10)

For -8,

x =-8+3 = -5, then y is (7+3x)/4 = (7-15)/4 = -2

Point is (-5,-2)