A rectangular hyperbola has parametric equations x = 4t, y = 4/t , (z non 0). Points P and Q on this hyperbola have parameters t = 1/4 and t = 2. Find the equation of the line l which passes through the origin and is perpendicular to the line PQ.
Since we are told that the line l is perpendicular to the line PQ, we first need to find out the gradient of PQ.
To do this, it is easiest to find out the co-ordinates of points P and Q in terms of their cartesian co-ordinates.
At P, t = 1/4, and since P is a point on our hyperbola, we can substitute this into the equation x = 4t to see that x = 4 * 1/4 = 1. Similarly, y = 4 / t = 4 / (1/4) = 16. Therefore, P is at (1, 16). We use exactly the same method for point Q, this time using t = 2 to obtain the co-ordinates for Q (8, 2).
We can then calculate the gradient of PQ by using the formula gradient = (y2 - y1) / (x2 - x1) = (2 - 16) / (8 - 1) = -2
Hence the gradient of our line l is -1 / -2 = 1/2.
Since the question tells us that our line goes through the origin, we know that our intercept value must be zero, so the equation of our line is y = mx + c = 1/2x + 0 = 1/2 x
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