To solve this problem, we need to find the value of t in terms of k that makes the tangent lines perpendicular, find the slopes of those tangent lines, and finally find the coordinates of their point of intersection.
a) To determine when two lines are perpendicular, we can check if the product of their slopes is equal to -1. So, let's find the slopes of the tangent lines at (t, F(t)) and (-t, F(-t)) first.
To find the slope of a tangent line, we need to find the derivative of the function F(x). Taking the derivative of F(x) = kx^2 + 3 gives us:
F'(x) = 2kx
Now, let's find the slope of the tangent line at (t, F(t)):
F'(t) = 2kt
The slope of a line perpendicular to this tangent line will be the negative reciprocal of its slope:
Slope of perpendicular line = -1 / (2kt)
Similarly, let's find the slope of the tangent line at (-t, F(-t)):
F'(-t) = 2k(-t) = -2kt
Again, the slope of a line perpendicular to this tangent line will be the negative reciprocal:
Slope of perpendicular line = -1 / (-2kt) = 1 / (2kt)
Now, we can set up the equation for perpendicular lines:
Product of slopes = (slope of first line) * (slope of second line) = -1
(-1 / (2kt)) * (1 / (2kt)) = -1
Simplifying the equation:
1 / (4k^2t^2) = -1
Multiplying both sides by 4k^2t^2:
1 = -4k^2t^2
Dividing both sides by -4k^2:
-1/4k^2 = t^2
Take the square root of both sides:
t = ± sqrt(-1/4k^2) = ± i * (1 / (2k))
Therefore, t = ± i / (2k).
b) Let's find the slopes of the tangent lines mentioned in part (a). We already determined the slope of the tangent line at (t, F(t)) to be 2kt. The slope of the tangent line at (-t, F(-t)) is -2kt.
c) Finally, let's find the coordinates of the point of intersection of the tangent lines mentioned in part (a).
To find this point, we can substitute the x-coordinate (t) into the original function F(x).
F(t) = k(t^2) + 3
And substitute the x-coordinate (-t) into the original function F(x).
F(-t) = k((-t)^2) + 3 = k(t^2) + 3
Since both F(t) and F(-t) are equal, the y-coordinates of the point of intersection are the same. We can set the two equations equal to each other:
k(t^2) + 3 = k(t^2) + 3
The constants cancel out:
0 = 0
This equation is always true regardless of the value of t. Therefore, the coordinates of the point of intersection are the same for all values of t.