First question:
If we have a curve and external point (point that does not belong to that curve), how many tangents from that point can be drawn to the curve? For example, if we have curve $y = x^2$ and point $(2,3)$, there are 2 tangents from $(2,3)$: $y = 2x-1$ and $y = 6x -9 $.
Generally, can there be more tangents e.g. 3,4,… from some external point to some curve?
Second question:
Can tangent drawn at some point on the curve cut the curve somewhere else?
I would appreciate any help.
Best Answer
For the first question
There can be infinitely many tangents to a curve (not always ofcourse). For eg. consider $y=\sin x$ and the line $y=1$. It touches the curve at infinitely many points. For the fig given assume the external pt. is $(4,1)$. Then, the $y=1$ is a tangent to infinite number of points on the $\sin x$ curve through that point.
For the second question
Yes, a tangent at a point can cut the curve somewhere else. Let,$y=x\sin x$. Each tangent for a local maxima cuts the curve somewhere else.
Interesting read: A tangent at a point of inflection can cut right through the curve rather than just touching it. Do some research on it, if interested.
For eg.