Line passing through a point and normal to a curve

That is correct. In summary, the equation of a straight line passing through (1,2) and normal to ##~x^2=4y## is ##~x+y=3##.
  • #1
Karol
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Homework Statement


What is the equation of a straight line passing through (1,2) and normal to ##~x^2=4y##

Homework Equations


Slopes of perpendicular lines:
$$m_2=-\frac{1}{m_1}$$

The Attempt at a Solution


$$x^2=4y~\rightarrow~m_1=y'=\frac{x}{2}$$
$$m_2=-\frac{2}{x}$$
$$2=-\frac{2}{1}+b~\rightarrow~b=4$$
The equation of the line: ##~y=-2x+4##
The answer should be ##~x+y=3##
 
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  • #2
The x you use in m2 = -2/x is not the x value at the (1,2) point. It needs to be the x of the (x,y) values where the normal line intersects the equation.

In the given answer, the intersection is at (2,1) and that gives m2 = -1. That agrees with m1 = 1 = slope of y=x2/4 at (2,1). y' = x/2 = 1.
 
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  • #3
If you use the point slope form of a line: y - y0 = m*(x - x0) then you can solve it. Assign (x0,y0) the point (1,2), and then (x,y) is a point on the parabola. Substitute y = x2 / 4, and then m = -2/x. You should then be able to solve for x.
 
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  • #4
$$\frac{y_1-y_2}{x_1-x_2}=\frac{\frac{x^2}{4}-2}{x-1}=-\frac{2}{x}$$
$$\rightarrow~x^3=16~\rightarrow~x=2$$
$$\rightarrow~y=1$$
$$\rightarrow~y+x=3$$
Thank you Ray and Scott
 
  • #5
You mean x3 = 8 → x=2.
 

Related to Line passing through a point and normal to a curve

What is a line passing through a point and normal to a curve?

A line passing through a point and normal to a curve is a line that intersects the curve at a right angle at a specific point, called the point of tangency.

How is the equation of a line passing through a point and normal to a curve determined?

The equation of a line passing through a point and normal to a curve can be determined using the point-slope form, where the slope is the negative reciprocal of the derivative of the curve at the point of tangency.

What is the significance of a line passing through a point and normal to a curve?

A line passing through a point and normal to a curve can help determine the direction of the curve at that specific point, as well as the rate of change of the curve. It is also useful in finding the equation of the tangent line, which can be used to approximate the curve at that point.

How does the slope of the curve at the point of tangency relate to the slope of the line passing through the point and normal to the curve?

The slope of the curve at the point of tangency is equal to the negative reciprocal of the slope of the line passing through the point and normal to the curve. This means that the two slopes are perpendicular to each other.

Can there be multiple lines passing through a point and normal to a curve?

No, there can only be one line passing through a point and normal to a curve at a specific point. However, there can be multiple points of tangency on a curve, each with its own unique line passing through it.

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