Angular Coeficient of a Line Passing Through (4,3) in 1st Quadrant with Area 27

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SUMMARY

The angular coefficient of a straight line passing through the point (4,3) in the first quadrant, which forms a triangle with the coordinate axes and has an area of 27 square units, can be determined using the equation of the line y = aX + B. By calculating the x and y intercepts in terms of the slope 'a', and setting the area formula (Area = 1/2 * base * height) equal to 27, the solutions for 'a' are found to be -1.5 and -0.375.

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Homework Statement


What is the angular coeficient of a straight line that passes through (4,3), if it satifies x>0 and y>0, and it is one of the sides of a triangle with total area of 27 (units of area).

calcule o coeficiente angular de um reta que passe por (4,3) de modo que a parte da reta no primeiro quadrante forma um triângulo de área 27 com os eixos coordenados positivos.


Homework Equations


I know that the area of a triangle is area = base * height/2 . I know the equation of the straigh line is y = aX + B, where "a" is what I am looking for. But I am stucked! I can't find a nice relation.



The Attempt at a Solution



I've tried to use the relation Area = 1/2 * D
Where D is the determinant of a matrix with the 3 vertices of the triangle, with an aditional columns of "1". But, couldn't go any further... any ideas?
 
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amarante said:

Homework Statement


What is the angular coeficient of a straight line that passes through (4,3), if it satifies x>0 and y>0, and it is one of the sides of a triangle with total area of 27 (units of area).

calcule o coeficiente angular de um reta que passe por (4,3) de modo que a parte da reta no primeiro quadrante forma um triângulo de área 27 com os eixos coordenados positivos.


Homework Equations


I know that the area of a triangle is area = base * height/2 . I know the equation of the straigh line is y = aX + B, where "a" is what I am looking for. But I am stucked! I can't find a nice relation.



The Attempt at a Solution



I've tried to use the relation Area = 1/2 * D
Where D is the determinant of a matrix with the 3 vertices of the triangle, with an aditional columns of "1". But, couldn't go any further... any ideas?

I assume the axes form the other two sides of the triangle. Write the equation of the line though ##(4,3)## with slope ##a##. Its ##x## and ##y## intercepts will give you the base and height in terms of ##a##. Then set the area, now in terms of ##a##, equal to 27 and solve for ##a##.
 
Perfect! I got it. Thank you so much!

I got a solution with two possibilities, a= -1.5, or a = -0.375.

Thanks!
 

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