Question

The coordinates of the point of intersection of two tangents to a rectangular hyperbola referred to its asymptote as axes are-

• A. Geometric means between the coordinates of the point of contact
• B. Arithmetic means between the coordinates of the point of contact
• C. Harmonic means between the coordinates of the point of contact
• D. None of the above

Answer 1

Answer: (C)

Harmonic means between the coordinates of the point of contact

Answer 2

The equation of the hyperbola referred to its asymptotes as the coordinates axes is xy = c².

The equations of the tangents to it at

P $\left( ct_{1},\frac{c}{t_{1}} \right)$and Q $\left( ct_{2},\frac{c}{t_{2}} \right)$ are $\frac{x}{t_{1}} + yt_{1}$= 2c

And $\frac{x}{t_{2}}$ + yt₂ = 2c.

These two intersect at R $\left( \frac{2ct_{1}t_{2}}{t_{1} + t_{2}},\frac{2c}{t_{1} + t_{2}} \right)$

Now, H.M. of ct₁ and ct₂ is $\frac{2ct_{1}.ct_{2}}{ct_{1} + ct_{2}}$ = $\frac{2ct_{1}t_{2}}{t_{1} + t_{2}}$

and, H.M. of $\frac{c}{t_{1}}$ and $\frac{c}{t_{2}}$ is $\frac{2\frac{c}{t_{1}}.\frac{c}{t_{2}}}{\frac{c}{t_{1}} + \frac{c}{t_{2}}}$ = $\frac{2c}{t_{1} + t_{2}}$.

Hence the coordinates of R are the harmonic means of the coordinates of the points of contact P and Q.