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A =(10,-10) and O=(0,0) Determine the sum of all x and y coordinates of all points Q on the line y =x + 8 such that angle OQA =90

 May 23, 2022
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It is given that angle OQA = 90 degrees, which means \((\text{slope of }OQ) (\text{slope of }QA) = -1\).

 

Let Q = (x, x + 8). Then \(\dfrac{(x + 8) - 0}{x - 0}\cdot \dfrac{(x + 8) - (-10)}{x - 10} = -1\)

\((x + 8)(x + 18) + x(x - 10) = 0\)

If you expand and solve the equation, you will find that this equation has no real solutions.

It turns out that the line y = x + 8 is too far away from O and A, so there are no such point on Q such that angle OQA = 90 degrees. Even the point on y = x + 8 closest to the midpoint of OA gives an acute angle OQA.

 May 23, 2022

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