Question

Normals at three points $P,Q,R$ on a rectangular hyperbola, intersect at a point on the curve; the centroid of the triangle $PQR$ is

• A. the centre of the hyperbola
• B. a focus of the hyperbola
• C. an extremity of a latus rectum
• D. none of these

Answer 1

The correct option is A the centre of the hyperbola

Let the equation of the rectangular hyperbola be
$xy=c^2$ ...(1)
The equation of the normal t any point 't' in (1) is
$x{t}^3-yt-c{t}^4+c=0$ ...(2)
If (2) passes through the point $S(c{t}^{\prime },\frac{c}{t^{\prime }})$ on hyperbola (1).
we have
$t^3{t^{\prime }}^2-t-t^4{t}^{\prime }+t^{\prime }=0$

$t^3{t}^{\prime }(t^{\prime }-t)+t^{\prime }-t=0$

$t^3{t}^{\prime }+1=0$, ----(3) Since $t^{\prime }\ne t$

the equation (3) is a cubic in $t$ showing that there are three points on the hyperbola (1)

the normals at which pass through the point $S$ on (1). Let the three values of $t$ given by (3) be $t_1,t_2,t_3$.
There are then the values of parameter $t$ at the point $P,Q,R$.
By theory of equation, we have
$t_1+t_2+t_3=\sum t_1=0$,
$\sum t_1{t}_2=0$,
$t_1{t}_2{t}_3=-1$
Let$(\overline{x},\overline{y})$ be the centroid of the $△PQR$.
Then $\overline{x}=\frac{1}{3}(c{t}_1+c{t}_2+c{t}_3)=0$
$\overline{y}=\frac{1}{3}(\frac{c}{t_1}+\frac{c}{t_2}+\frac{c}{t_3})=\frac{c}{3}.\frac{t_2{t}_3+t_1{t}_3+t_1{t}_3}{t_1{t}_2{t}_3}=0$
Thus, the centroid of the $△PQR$ is $(0,0)$ which is centre of the hyperbola.