Geometria no Plano e no Espaço - Exercício 5

Considere, num referencial ortonormado do plano, retas r e s definidas por:

r:{\text{  }}\left( {x,y} \right) = \left( {1, - 3} \right) + k\left( {2,3} \right),k \in \mathbb{R}

s:{\text{  }}y = 4x - 1

Determine as coordenadas do ponto de interseção das retas r e s.

 

 

Resolução do exercício de matemática:

 

  • Escrever a equação reduzida da reta r:

 

m = \frac{3}{2}

 

A equação reduzida da reta é do tipo:  y = \frac{3}{2}x + b.

 

Como P\left( {1, - 3} \right) é um ponto da reta r, tem-se que:

 

 - 3 = \frac{3}{2} \times 1 + b \Leftrightarrow b =  - \frac{9}{2}

 

Logo, a equação reduzida da reta r é:  y = \frac{3}{2}x - \frac{9}{2}.

 

  • Coordenadas do ponto de interseção das retas r e s:


\left\{ {\begin{array}{*{20}{c}} {y = \frac{3}{2}x - \frac{9}{2}} \\ {y = 4x - 1} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {y = \frac{3}{2}x - \frac{9}{2}} \\ {\frac{3}{2}x - \frac{9}{2} = 4x - 1} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {y = \frac{3}{2}x - \frac{9}{2}} \\ {\frac{3}{2}x - 4x = - 1 + \frac{9}{2}} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {y = \frac{3}{2}x - \frac{9}{2}} \\ {\frac{3}{2}x - \frac{8}{2}x = - \frac{2}{2} + \frac{9}{2}} \end{array}} \right. \Leftrightarrow

 

 \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {y = \frac{3}{2}x - \frac{9}{2}} \\ {3x - 8x = - 2 + 9} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {y = \frac{3}{2}x - \frac{9}{2}} \\ { - 5x = 7} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {y = \frac{3}{2}x - \frac{9}{2}} \\ {x = - \frac{7}{5}} \end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {y = \frac{3}{2}\left( { - \frac{7}{5}} \right) - \frac{9}{2}} \\ {x = - \frac{7}{5}} \end{array}} \right. \Leftrightarrow

 

 \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {y = - \frac{{21}}{{10}} - \frac{9}{2}} \\ {x = - \frac{7}{5}} \end{array} \Leftrightarrow } \right.\left\{ {\begin{array}{*{20}{c}} {y = - \frac{{21}}{{10}} - \frac{{45}}{{10}}} \\ {x = - \frac{7}{5}} \end{array} \Leftrightarrow } \right.\left\{ {\begin{array}{*{20}{c}} {y = - \frac{{76}}{{10}}} \\ {x = - \frac{7}{5}} \end{array} \Leftrightarrow } \right.\left\{ {\begin{array}{*{20}{c}} {y = - \frac{{38}}{5}} \\ {x = - \frac{7}{5}} \end{array}} \right.

 

Logo, I\left( { - \frac{7}{5}, - \frac{{38}}{5}} \right).