Question

Find the coordinates of all points on the graph of y=23−x2 at which the tangent line passes through the point (10,26).please show the work i greatly appreciate it :)

Answer 1

y = 23 - x²
y' = -2x

So, we want all points where the line through (x,y) and (10,26) has slope -2x

(y-26)/(x-10) = -2x
(23-x² - 26)/(x-10) = -2x
-3 - x² = -2x(x-10)
-3 - x² = -2x² + 20x
x² - 20x - 3 = 0
x = -0.149 or 20.149

At those values for x, we have
(x,y) = (-0.149,22.978) and (20.149,-382.982)

y' = 0.290 and -40.298

the line through (-0.149,22.978) and (10,26) has slope 3.022/10.149 = 0.29

the line through (20.149,-382.982) and (10,26) has slope 408.982/-10.149 = 40.298

Look slike those are our lines.