coordinate geometry

Question

the point A has coordinates (-2,5) and the point B has coordinates (8,-6).
a) find an equation for the straight line AB, giving your answer in the from px+qy=r, where p,q, and r are integers.
b) the point C has coordinates (k,k+1). given that angle ACB is a right angle find the two possible values of k.

 

i got k=38/21 and k=-135 but i feel like that's wrong can someone do part B step by step?

Answer 1

Using slopes of perpendicular lines.

 

m_AC = (k + 1 - 5) / (k + 2) ⇒ m_AC = (k - 4) / (k + 2)

m_BC = (k + 1 + 6) / (k - 8) ⇒ m_BC = (k + 7) / (k - 8)

 

To be perpendicular

m_AC = -1 / m_BC

(k - 4) / (k + 2) = -(k - 8) / (k + 7)

(k - 4)(k + 7) = -1(k - 8)(k + 2)

k² + 3k - 28 = -(k² - 6k - 16)

k² + 3k - 28 = -k² + 6k + 16

2k² - 3k - 48 = 0 

 

Unless the arithmetic is in error (a possibility) the roots are irrational.