The points \(A(1,\,-2)\), \(B(3,\,k)\) and \(C(7,\,10)\) lie on one straight line. What is the value of \(k\)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(2\)
Three points lie on one line when the area of the triangle they form is zero: \(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)=0\).
\(1(k-10)+3(10-(-2))+7(-2-k)=0\)
\(k-10+36-14-7k=0\)
\(12-6k=0\)
\(k=2\)
Check: from A to B, x goes up by 2 and y by 4; from B to C, x goes up by 4 and y by 8, the same rate, so A, B, C are on one line
