A rectangular hyperbola- Math Central

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Question from Gareth, a student:

2. Consider the rectangular hyperbola xy = c^2 with parameterization (x, y) = (ct, c/t), and t 6= 0.
i. Derive (i.e. do not just quote) the equations of the tangent and normal at the point P with
(parametric) coordinates (cp, c/p).
ii. Consider the points P : (cp, c/p) and Q : (cq, c/q) on the hyperbola. Find the equation of the
straight line that joins the points P and Q.
iii. Consider a point R : (cr, c/r) also on the hyperbola, and suppose that in the triangle PRQ,
\PRQ = 90◦. Prove that the normal (to the hyperbola) at R is parallel to the line PQ.

Gareth,

The equation of the hyperbola is y = c²/x = c² x^-1 and hence dy/dx = - c² x^-2 = - c²/x². Thus at P where x = cp, the slope of the tangent line is - c²/(cp)² = - 1/p². The tangent line at P is hence the line through (cp, c/p) and slope - 1/p². The normal line at P is then the line trough (cp, c/p) with slope p².
Write the equation of the line through P (cp, c/p) and Q (cq, c/q).
In ii. you found the slope of the line joining P and Q so you need to show that the normal at R has the same slope. From i. you know the slope of the normal at R but this is in terms of r where the slope of the line joining P and Q is in terms of p and q. Write m₁, the slope of the line joining P and R and m₂, the slope of the line joining Q and R. Since these lines are perpendicular, the slopes must be negative reciprocals. This will give you a relationship between m₁ and m₂ which will allow you to write the slope of the normal at R in terms of p and q.

I hope this helps,
Harley

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