Suppose that all the tangent lines of a regular plane curve pass through some fixed point. Prove that the curve is part of a straight line. Prove the same result if all the normal lines are parallel.
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To prove that the curve is part of a straight line when all the tangent lines pass through a fixed point, we can use the concept of the curve's curvature.
Let's assume that the curve is not a straight line. If it is not a straight line, then it must have some curvature at every point.
Now, consider a point on the curve and draw the tangent line at that point. According to the given condition, this tangent line must pass through the fixed point.
Since the curve has curvature at every point, the tangent line will always deviate from being parallel to the fixed point. This implies that the tangent lines cannot all pass through the fixed point, contradicting the given condition.
Therefore, we can conclude that the curve must be part of a straight line if all the tangent lines pass through a fixed point.
Similarly, let's prove that if all the normal lines are parallel, then the curve is part of a straight line.
The normal line at any point on a curve is perpendicular to the tangent line at that point. So, if all the normal lines are parallel, it means that all the tangent lines are also parallel.
Let's assume that the curve is not a straight line. If it is not a straight line, then there must be some curvature at every point on the curve.
Now, consider two points on the curve that are close to each other. Since the curve has curvature, the tangent lines at these two points will not be parallel. But we assumed all the tangent lines are parallel, which is a contradiction.
Therefore, our assumption that the curve is not a straight line must be false. Hence, if all the normal lines are parallel, then the curve is part of a straight line.
So, in both cases, the given conditions imply that the curve is part of a straight line.