# Analytic Geometry World Problem

• S.R
In summary, the problem involves finding coordinate pairs for point T, relative to point S from the left, on a line with equation 2x+y=3, given that the distance between points S and T is 13 units. The approach used involves setting up a system of equations and using the distance formula to find the solutions, which can also be obtained by considering a circle with a diameter of 13 at point (2,-1).
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?

Last edited:
S.R said:

## 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...

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
[tex](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.

HallsofIvy said:
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
[tex](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.

Thanks HallsofIvy!

## What is Analytic Geometry World Problem?

Analytic Geometry World Problem is a branch of mathematics that combines algebra and geometry to solve real-world problems. It involves using coordinates and equations to describe geometric shapes and their properties.

## How is Analytic Geometry World Problem used in real life?

Analytic Geometry World Problem is used in a variety of fields, such as engineering, physics, and architecture. It helps in the design and analysis of structures, as well as in the study of motion and forces.

## What are the key concepts in Analytic Geometry World Problem?

The key concepts in Analytic Geometry World Problem include points, lines, planes, distance, slope, and equations of geometric shapes such as circles, ellipses, and parabolas.

## What are some common problem-solving strategies in Analytic Geometry World Problem?

Some common problem-solving strategies in Analytic Geometry World Problem include plotting points on a coordinate plane, using equations to find intersecting points, and using geometric formulas to calculate distances and angles.

## How can I improve my skills in Analytic Geometry World Problem?

To improve your skills in Analytic Geometry World Problem, you can practice solving a variety of problems, review key concepts and formulas, and seek help from teachers or tutors if needed. Additionally, you can use online resources and tools to visualize and solve problems.

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