The difference between the x-coordinates of two points on the parabola y^2=4ax is fixed at 2k. Find the equation that describes the position(xp, py)of the point of intersection P of the tangents at the two points. The equation is in the form yp^2=f(xp).
Hence show that 4axp<yp^2<4axp+2k.
To find the equation that describes the position (xp, py) of the point of intersection P of the tangents at the two points on the parabola y^2 = 4ax, we need to follow these steps:
Step 1: Determine the points of tangency
First, we need to find the points of tangency on the parabola. Let's assume the two points of tangency are (x1, y1) and (x2, y2).
Given that the difference between the x-coordinates of the two points is fixed at 2k, we have:
x1 - x2 = 2k
Step 2: Find the equations of the tangents at these points
Next, we calculate the equations of the tangents at points (x1, y1) and (x2, y2).
The slope of the tangent at (x, y) on the parabola y^2 = 4ax is given by:
dy/dx = 2a/y
Using the point-slope form, the equation of the tangent at (x1, y1) is:
(y - y1) = (2a/y1) * (x - x1)
Similarly, the equation of the tangent at (x2, y2) is:
(y - y2) = (2a/y2) * (x - x2)
Step 3: Find the point of intersection P of the tangents
To find the coordinates (xp, py) of the point of intersection P, we solve the simultaneous equations of the two tangent lines obtained in Step 2.
Substitute the equations of the tangents into each other to eliminate y:
(2a/y1) * (x - x1) = (2a/y2) * (x - x2)
Cross-multiplying and simplifying, we get:
y1*(x - x1) = y2*(x - x2)
Expanding the equation further:
yx - y1*x1 = yx - y2*x2
Canceling yx from both sides, we get:
y1*x1 = y2*x2
Step 4: Simplify the equation and express in the given form
Now, using the relation y^2 = 4ax, we substitute y1 = √(4ax1) and y2 = √(4ax2) into the equation obtained in Step 3.
(√(4ax1)) * x1 = (√(4ax2)) * x2
Simplifying and rearranging, we get:
4ax1*x1 = 4ax2*x2
Dividing by 4a, we have:
x1*x1 = x2*x2
Taking the square root of both sides, we get:
x1 = ±x2
Since the difference between x1 and x2 is 2k, we can substitute x1 = x2 + 2k or x1 = x2 - 2k into the equation.
Step 5: Express yp^2 in terms of xp and show the inequality
Substituting x1 = x2 + 2k into the equation x1 = ±x2, we get:
x2 + 2k = ±x2
Simplifying, we have:
±2k = -x2
The negative case (-2k = -x2) is not possible since the difference between x1 and x2 is given as positive 2k. Therefore, we take the positive case (2k = -x2).
Substituting this into the equation y^2 = 4ax, we have:
yp^2 = 4a(-x2)
Simplifying further, we get:
yp^2 = -4ax2
Since x1 = x2 + 2k, we have x2 = x1 - 2k. Substituting this into the equation:
yp^2 = -4a(x1 - 2k)
Expanding, we get:
yp^2 = -4ax1 + 8ak
Now, we need to show that 4axp < yp^2 < 4axp + 2k.
First, let's consider 4axp < yp^2:
Since x1 = xp + k, we substitute this into the equation above:
4a(xp + k) < yp^2
Expanding, we get:
4axp + 4ak < yp^2
Now, let's consider yp^2 < 4axp + 2k:
Since x1 = xp - k, we substitute this into the equation above:
yp^2 < 4a(xp - k) + 2k
Expanding, we get:
yp^2 < 4axp - 4ak + 2k
Combining the two inequalities, we get the desired result:
4axp + 4ak < yp^2 < 4axp - 4ak + 2k
Simplifying further, we have:
4axp < yp^2 < 4axp + 2k
Therefore, we have shown that 4axp < yp^2 < 4axp + 2k, as required.