First slide

Hyperbola in conic sections

Question

If the normals at $P, Q, R$ on the rectangular hyperbola $x y = c^{2}$ intersect at a point S on the hyperbola, then centroid of the triangle $P Q R$ is at

Moderate

A

an extremity of the latus rectum
B

the centre
C

a focus
D

the point $S$ on the hyperbola

Solution

Normal at any point $(c t, c / t)$ on the hyperbola $x y = c^{2}$ if $y - (c / t) = t^{2} (x - c t) or t^{3} x - t y + c - c t^{4} = 0$ If it passes through another points $(c α, c / α)$ on the hyperbola, then
$t^{3} \times c α - t \times \frac{c}{α} + c - c t^{4} = 0$
$\begin{array}{l} \Rightarrow t^{3} α^{2} - t + α - α t^{4} = 0 \\ \Rightarrow (t^{3} α + 1) (α - t) = 0 \\ \Rightarrow t^{3} α + 1 = 0 as α \neq t \end{array}$
which gives three values of t say $t_{1}, t_{2}, t_{3}$ and hence three points P, Q, R on the hyperbola the normals at which pass through S.
We have $t_{1} + t_{2} + t_{3} = 0 = Σ t_{1} t_{2} and t_{1} t_{2} t_{3} = - 1 / α$
Centroid of the triangle PQR is
$(\frac{c (t_{1} + t_{2} + t_{3})}{3}, \frac{c (1 / t_{1} + 1 / t_{2} + 1 / t_{3})}{3}) = (0, 0)$
which is the centre of hyperbola.

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