Is the Point (-1/2, -2) Equidistant from Two Perpendicular Lines?

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Homework Help Overview

The discussion revolves around determining whether the point (-1/2, -2) is equidistant from the lines defined by the equations 2x - 3y + 4 = 0 and 6x + 4y - 7 = 0. The problem involves concepts from geometry related to distances from points to lines, particularly in the context of perpendicular lines.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relevance of the distance formula and the implications of the lines being perpendicular. Some suggest calculating distances from the point to each line to compare them. Others question the necessity of certain calculations and explore the geometric relationships between the lines and the point.

Discussion Status

The discussion includes various perspectives on how to approach the problem. Some participants provide hints and suggest methods for calculating distances, while others express uncertainty about the relevance of certain equations and concepts. There is no explicit consensus, but several productive lines of reasoning are being explored.

Contextual Notes

Some participants note that the lines are not parallel and discuss the implications of their perpendicularity on the distances involved. There is mention of a potential trick to simplify the calculations due to the geometric properties of the lines.

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Homework Statement


Show that the point (-1/2, -2) is equidistant from the straight lines 2x-3y+4=0 and 6x+4y-7=0


Homework Equations



distance between the points= {(x1-x2)^2+(y1-y2)^2}^1/2

The Attempt at a Solution


All I know is that the given two straight lines are perpendicular to each other. It would be very helpful if anyone of you could drop a hint as to how I should use this fact in proving the fact that the given point is at an equal distance from the given straight lines
 
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Did you draw any graphs?
 
I don't see the relevance of your equation.
Do you know how to calculate the distance between two parallel lines ?

For example between 6x + 4y = 7 and 6x + 4y = -11 ?

(The latter happens to go through the point -1/2, -2 )
 
BvU said:
I don't see the relevance of your equation.
Do you know how to calculate the distance between two parallel lines ?

For example between 6x + 4y = 7 and 6x + 4y = -11 ?

(The latter happens to go through the point -1/2, -2 )

I don't see the relevance of your questions. The two lines given by the OP are not parallel.
 
Dumbledore211 said:

Homework Statement


Show that the point (-1/2, -2) is equidistant from the straight lines 2x-3y+4=0 and 6x+4y-7=0


Homework Equations



distance between the points= {(x1-x2)^2+(y1-y2)^2}^1/2

The Attempt at a Solution


All I know is that the given two straight lines are perpendicular to each other. It would be very helpful if anyone of you could drop a hint as to how I should use this fact in proving the fact that the given point is at an equal distance from the given straight lines

Just find the distance between the given point and the first line .Then calculate the distance between the point and the second line .You will get same distances in the two cases .

The distance between a point (p,q) and line Ax+By+C=0 is given by |Ap+Bq+C|/√(A2+B2)
 
sorry post error
 
Ray Vickson said:
I don't see the relevance of your questions. The two lines given by the OP are not parallel.

No, but because the lines are perpendicular to each other, the 'dropped perpendicular' from the point to one line will be parallel to the other line. Whether that saves anything I'm not sure.
 
Dear professor Dumbledore,
Do you recognize/understand the formula as spelled out by Tanya? If so, the exercise is straightforward as she says (work, yuch!).
In #1 (2) you made me a bit careful, since there are no two points in the OP. Yet in #1 (3) you quite correctly observe the two lines are perpendicular and you sniff that there might be a smart mathemagical trick to avoid superfluous work. Good attitude!
 

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