Locating a Moving Point P: x+y=9

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In summary, the problem involves a moving point P that is tangent to the circle x2+y2=16 and is a distance of 8 from the point (8,8). The goal is to show that the locus of P is the straight line x+y=9. This can be solved by equating the length of the tangent to the distance from (8,8) and solving for the coordinates of P.
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Michael_Light
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Homework Statement



A moving point P is such that the length of the tangent from P to the circle x2+y2=16 is equal to the distance of P from point (8,8). Show that the locus of P is the straight line x+y=9.

Homework Equations





The Attempt at a Solution



I sketched a graph for this, but it doesn't seems to help me a lot in solving this question. Can anyone give me some hints? Thanks...
 
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Well, let the point P be (a,b). What is the length of the tangent to the circle? What is its distance from (8,8)? Equate the two.
 

1. How do you locate a moving point P on the graph of x+y=9?

The equation x+y=9 represents a straight line with a slope of -1 and a y-intercept of 9. To locate a moving point P on this line, you would need to know its coordinates (x,y). This can be done by plugging in different values for x and solving for y, or vice versa. The resulting points can be plotted on the graph to show the movement of point P.

2. What is the significance of the equation x+y=9 in locating a moving point P?

The equation x+y=9 is significant because it represents a relationship between the x and y coordinates of any point on the graph. This relationship helps us understand the movement and position of a point P on the graph.

3. How does the slope of the line in x+y=9 affect the movement of point P?

The slope of the line in x+y=9 is -1, which means that for every increase of 1 unit in the x direction, there will be a decrease of 1 unit in the y direction. This affects the movement of point P by determining its direction and rate of change along the line.

4. Can you use the equation x+y=9 to predict the future position of point P?

Yes, you can use the equation x+y=9 to predict the future position of point P. By plugging in different values for x or y, you can determine the coordinates of the point at a specific time or distance. This can help you understand and predict the movement of point P on the graph.

5. Are there any other methods besides using the equation x+y=9 to locate a moving point P?

Yes, there are other methods to locate a moving point P on the graph of x+y=9. These include using the slope-intercept form, point-slope form, or graphing the equation on a coordinate plane. Each method can provide different insights into the movement and position of point P.

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