Question

A tangent line is drawn to the hyperbola \(xy = c\) at a point P as shown in the figure.

(a) Show that the mid-point of the line segment cut from this tangent line by the coordinate axes is P.

(b) Show that the triangle formed by the tangent line and the corodiante axes always has the same area, no matter where P is located on the hyperbola.

Short answer

(a) The coordinate of the mid-point are \(\left( {a,\frac{c}{a}} \right)\).

(b) It is proved that the area of the triangle is constant, for tangent at any point.

Step-by-step solution

Find the answer for part (a)

(a) The coordinate of a point on the curve \(xy = c\) is \(P\left( {a,\frac{c}{a}} \right)\). The value of \(\frac{{{\rm{d}}y}}{{{\rm{d}}x}}\) can be calculated as,

\(\begin{aligned}\frac{{{\rm{d}}y}}{{{\rm{d}}x}} &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{c}{x}} \right)\\ &= - \frac{c}{{{x^2}}}\end{aligned}\)

So, the value of \(y'\left( a \right)\) is:

\(y'\left( a \right) = - \frac{c}{{{a^2}}}\)

Find the equation of tangent line is \(x = a\)

The equation of tangent at point \(\left( {a,\frac{c}{a}} \right)\) with slope \( - \frac{c}{{{a^2}}}\) can be calculated as,

\(\begin{aligned}y - \frac{c}{a} &= - \frac{c}{{{a^2}}}\left( {x - a} \right)\\y &= - \frac{c}{{{a^2}}}x + \frac{{2c}}{a}\end{aligned}\)

Find the mid-point of the tangent line between the axes

The line \(y = - \frac{c}{{{a^2}}}x + \frac{{2c}}{a}\) intersect the axes at \(\left( {0,\frac{{2c}}{a}} \right)\)and \(\left( {2a,0} \right)\). The mid-point coordinate can be calculated as follows:

\(\begin{aligned}P &\equiv \left( {\frac{{0 + 2a}}{2},\frac{{\frac{{2c}}{a} + 0}}{2}} \right)\\ &\equiv \left( {a,\frac{c}{a}} \right)\end{aligned}\)

Thus, the coordinate of mid-point between the axes is \(\left( {a,\frac{c}{a}} \right)\).

Find the area of the triangle fromed by the tangent at P

(b) The coordainte of vertices of triangle from by the tangent are \(\left( {0,\frac{{2c}}{a}} \right)\), \(\left( {2a,0} \right)\) ,and \(\left( {0,0} \right)\). So, the area of the trangle is:

\(\begin{aligned}A &= \frac{1}{2}\left( {2a} \right)\left( {\frac{{2c}}{a}} \right)\\ &= 2c\end{aligned}\)

As c is a constant, therefore the area of the triangle made by the tangent is \(2c\).

It is proved that the triangle formed by the tangent line and the corodiante axes always has the same area, no matter where P is located on the hyperbola.