Analytic Geometry World Problem

[S.R]

Homework Statement

Points S and T lie on the line 2x+y=3. If the length ST is 13 units, and the coordinates of S are (2, -1), determine all possible coordinate pairs for T, correct to two decimal places.

For simplicity, determine the coordinates for T, relative to S from the left.

Homework Equations

a^2=b^2+c^2 (in this problem, the distance formula.)

The Attempt at a Solution

I used a system of equations:

y=-2x+3

(2-x)^2+(-1-y)^2=13^2

This system is quite tedious. I used WolframAlpha to compute the solutions, however, they did not match the answer in my textbook.

Is this approach correct?

 

[kingstrick]

Homework Statement

Points S and T lie on the line 2x+y=3. If the length ST is 13 units, and the coordinates of S are (2, -1), determine all possible coordinate pairs for T, correct to two decimal places.

For simplicity, determine the coordinates for T, relative to S from the left.

Homework Equations

a^2=b^2+c^2 (in this problem, the distance formula.)

The Attempt at a Solution

I used a system of equations:

y=-2x+3

(2-x)^2+(-1-y)^2=13^2

This system is quite tedious. I used WolframAlpha to compute the solutions, however, they did not match the answer in my textbook.

Is this approach correct?

Think of a circle with a diameter of 13 at point (2,-1). It helps if you actually draw this. To get to a coordinate or point on the arc you would...

 

[HallsofIvy]

Kingstrick, that is exactly what he is doing. S.R., yes, that is a valid method. of course, you can replace the y in the circle equation by -2x+ 3 so that it becomes
$$(2- x)^2+ (-1-(-2x+3))^2= (2- x)^2+ (2x- 4)^2= 13. Multiplying that out will give you a quadratic equation which will have 2 solutions because, of course, this line is a diameter of the circle and so crosses it twice.$$

 

[S.R]

Kingstrick, that is exactly what he is doing. S.R., yes, that is a valid method. of course, you can replace the y in the circle equation by -2x+ 3 so that it becomes
$$(2- x)^2+ (-1-(-2x+3))^2= (2- x)^2+ (2x- 4)^2= 13. Multiplying that out will give you a quadratic equation which will have 2 solutions because, of course, this line is a diameter of the circle and so crosses it twice.$$

$$<br /> Thanks HallsofIvy!$$