(1,0) is a point on the parabola y=2x(x−1). From point A, point P is moving along the curve towards the origin O(0,0). As P→O, sec^2∠APO→N, where N is a positive integer. What is the value of N?
To find the value of N, we need to determine the angle ∠APO and find the limit of sec²∠APO as P approaches the origin O.
Given that point (1,0) lies on the parabola y = 2x(x - 1), we can find the equation of the tangent line at A(1,0) using calculus.
1. Find the derivative of y with respect to x:
dy/dx = 2x(1) + 2x - 2 = 4x - 2
2. Evaluate the derivative at x = 1 (since A lies on the parabola):
dy/dx = 4(1) - 2 = 2
3. Using the point-slope form of a line, the equation of the tangent line is:
y - 0 = 2(x - 1)
y = 2x - 2
Now, let's find the equation of the line AP and the angle ∠APO.
4. The line AP passes through A(1,0) and approaches the origin O(0,0) as P→O.
5. Find the slope of line AP:
slope_AP = (0 - 0)/(1 - 0) = 0/1 = 0
6. Therefore, the equation of line AP is y = 0x + c, where c is the y-intercept.
7. To find c, substitute the coordinates of point A into the equation:
0 = 0(1) + c
c = 0
8. Thus, the equation of line AP is y = 0.
Since the tangent line and line AP are parallel, the angle ∠APO formed between them is 180 degrees.
Now, let's determine the limit of sec²∠APO as P approaches the origin O. As P approaches O, the distance between P and O becomes smaller and smaller.
To evaluate the limit, let P approach O along the line y = 0.
9. The distance between P and O along line y = 0 can be measured in the x-coordinate, which gives us the value of x.
10. As P approaches O, x approaches 0.
Now, let's find the limit of sec²∠APO as x approaches 0:
lim(x→0) sec²∠APO = sec²180°
The sec²(180°) is equal to 1/cos²(180°). The cosine of 180 degrees is -1, so
= 1/(-1)²
= 1/1
= 1
Therefore, the value of N is 1.