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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.

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user121
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1 Answers1

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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. enter image description here

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. enter image description here

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.enter image description here

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Soham
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    It's not clear from their wording whether OP considers the case of a single line being tangent to a curve at multiple points as separate tangents. If not, though, your second example $y = x \sin x$ is certainly an example. – Travis Willse Sep 23 '18 at 15:16
  • @Travis Yep. The wording is not clear. The OP deals with algebraic expressions while my I give examples of trigonometric expressions. It may not be as easy to give the same kind of examples using algebraic expressions. (Truth is I myself cannot think of an algebraic expression that has infinite number of tangents from an external point)... – Soham Sep 23 '18 at 15:19