Question

Three pointshave coordinates A( 7,4) B(19,8) and c(K,2K). Find the value of K constant for which C lies on the line that passes through the points A and B where angle CAB is right angle​

Answer 1

To find the value of K for which point C lies on line AB and creates a right angle at A, calculate the equation of line AB and ensure point C satisfies it. The result is that K must equal 1.

Step-by-step explanation:

To determine the value of K for which the point C lies on the line AB and forms a right angle at A, we need to both find the equation of the line AB and apply the condition for a right angle using the dot product of vectors. Point A has coordinates (7, 4) and point B has coordinates (19, 8). The slope of the line AB can be calculated as:

m = (8 - 4) / (19 - 7) = 4 / 12 = 1/3.

This means the equation of line AB is of the form y = (1/3)x + b. Substituting coordinates of point A to find b gives us 4 = (1/3)(7) + b, which leads to b = 4 - 7/3 = 5/3, so the equation of the line is y = (1/3)x + 5/3.

For point C with coordinates (K, 2K) to lie on the line y = (1/3)x + 5/3, it must satisfy this equation:

2K = (1/3)K + 5/3. Multiplying through by 3 to clear the fraction:

6K = K + 5 => 5K = 5 => K = 1.

Therefore, the value of K for which point C lies on the line AB and makes angle CAB a right angle is 1.