dustbin said:Homework Statement
The line that is normal to the curve x2+2xy-3y2=0 at (5,5) intersects the curve at what other point?
2. The attempt at a solution
I differentiated the equation, found the slope of the curve at that point, and I then found the equations for the tangent line and normal line. I'm not really sure where to approach from here... any hints?
A normal line is a line that is perpendicular to a curve at a specific point. It represents the instantaneous slope of the curve at that point.
Points of intersection are points where two lines, curves, or shapes intersect or cross each other. In the context of a normal line, points of intersection refer to the points where the normal line intersects with the curve.
The points that a curve's normal line intersects are calculated using calculus. Specifically, the derivative of the curve at a given point is used to find the slope of the normal line, and then this slope is used to find the equation of the normal line. The intersection points are then found by solving the equations of the curve and the normal line simultaneously.
The normal line is important in calculus because it helps us understand the behavior of a curve at a specific point. By finding the slope of the normal line, we can determine whether the curve is increasing or decreasing at that point, and by finding the equation of the normal line, we can calculate the rate of change of the curve at that point.
No, the normal line can only intersect the curve at a maximum of two points. This is because the normal line is always perpendicular to the curve, and two perpendicular lines can only intersect at one point. If the normal line intersects the curve at more than two points, it would no longer be perpendicular to the curve at those points.