Question 1191761
<br>
Here is a diagram with (arbitrarily) (a,b)=(3,8).<br>
{{{drawing(400,400,-4,20,-4,20
,line(-2,0,18,0),line(0,-2,0,18)
,graph(400,400,-4,20,-4,20,24/x)
,line(0,16,6,0),line(0,8,3,8),line(3,8,3,0)
,locate(4,9,"P(3,8)=(a,b)"),locate(6,1,Q),locate(.5,16.5,R)
)}}}<br>
The equation of the curve is<br>
{{{y=ab/x}}}<br>
The derivative of the function is<br>
{{{dy/dx=-ab/x^2}}}<br>
The slope of the curve at x=a is the derivative evaluated at x=a:<br>
{{{-ab/a^2=-b/a}}}<br>
At this point, to finish the problem, we could do some formal algebra with the point-slope equation of the tangent line to find the x- and y-intercepts and use the distance formula to show that PR=QR.<br>
However, with only a little thought about what we have here, we can see that with a point (3,8) and a tangent at that point with slope (-8/3), moving 3 to the left and 8 up gives us the y-intercept of the tangent line, and moving 3 to the right and down 8 gives us the x-intercept of the tangent line.  So the x- and y-intercepts are both the same distance from the point P in both the x- and y-directions, which means the lengths of PQ and PR are equal.<br>