Finding the X-coordinate of a normal to the tangent

In summary, the conversation is discussing a problem involving a curve with an equation y=x3-2x2-x+9 and a tangent at point P(2,7) with an equation of y=3x+1. The question asks to find the x-coordinate for a point Q on the curve that is perpendicular to the tangent. The attempt at a solution includes finding the gradient of the normal, but the point that the normal passes through has not been specified. The original question can be found in the provided link.
  • #1
Count Duckula
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Homework Statement



Tangent to C at point P(2,7) has an equation of y=3x+1.

Point Q also lies on C and is perpendicular to the tangent,
show that the X-coordinate is [1/3(2+√6)]

Homework Equations


curve C has equation y= x3-2x2-x+9
dy/dx = 3x2-4x-1

The Attempt at a Solution


gradient of normal has to = -1/3, right? considering that the tangent that is is perpendicular to is +3
I'm not sure as, the point that the normal is passing through hasn't been specified.

Here is the actual question: http://i259.photobucket.com/albums/hh299/the-real-guitar-hero/CapturePNGnnn_zpsd170ce09.png

Part C.
 
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  • #2
Count Duckula said:

Homework Statement



Tangent to C at point P(2,7) has an equation of y=3x+1.

Point Q also lies on C and is perpendicular to the tangent,
show that the X-coordinate is [1/3(2+√6)]

Homework Equations


curve C has equation y= x3-2x2-x+9
dy/dx = 3x2-4x-1

The Attempt at a Solution


gradient of normal has to = -1/3



I'm not sure as the point, that the normal is passing through hasn't been specified.

Here is the actual question: http://i259.photobucket.com/albums/hh299/the-real-guitar-hero/CapturePNGnnn_zpsd170ce09.png

Part C.

Problems involving derivatives should be posted in the Calculus & Beyond section. I am moving this thread to that section.
 

1. What is the definition of a normal to the tangent?

A normal to the tangent is a line that is perpendicular to the tangent line at a specific point on a curve. It is used to find the slope of the curve at that point, and can help determine the direction in which the curve is heading.

2. How do you find the x-coordinate of a normal to the tangent?

To find the x-coordinate of a normal to the tangent, you first need to find the derivative of the curve at the given point. This will give you the slope of the tangent line. Then, you can use the negative reciprocal of this slope to find the slope of the normal line. Finally, you can use the point-slope formula to find the equation of the normal line and solve for the x-coordinate.

3. What is the importance of finding the x-coordinate of a normal to the tangent?

Finding the x-coordinate of a normal to the tangent is important because it helps us understand the behavior of a curve at a specific point. It can also be used to find the equation of the tangent line, which is useful in many applications, such as optimization and curve fitting.

4. Are there any special cases when finding the x-coordinate of a normal to the tangent?

Yes, there are a few special cases when finding the x-coordinate of a normal to the tangent. For example, if the curve is a straight line, the normal line will be perpendicular and the x-coordinate will be the same as the given point. Another special case is when the slope of the tangent line is undefined, which can occur at a sharp turn or cusp in the curve.

5. Can I use calculus to find the x-coordinate of a normal to the tangent?

Yes, calculus is the most common method for finding the x-coordinate of a normal to the tangent. It relies on the concept of derivatives and slope to determine the equation of the tangent line and the normal line. However, there are also alternative methods, such as using the slope formula and the equation of a circle, which may be more suitable for certain situations.

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