Question

a distancia do ponto P (1.y) ate a reta s: x+y=0 é de raiz quadrada de 2/2 encontre o valor de y determine as coordenadas do ponto p

Answer 1

$P(1,\,y)\implies x_0=1,~y_0=y$

$s:x+y=0\implies a=1,~b=1,~c=0$

$\therefore$

      $d_{Ps}=\dfrac{ax_0+by_0+c}{\sqrt{a^2+b^2}}~~\Longleftrightarrow$

$\Longleftrightarrow ~~\dfrac{\sqrt{2}}{2}=\dfrac{1\cdot1+1\cdot y+0}{\sqrt{1^2+1^2}}~~\Longleftrightarrow$

$\Longleftrightarrow~~\dfrac{\sqrt{2}}{2}=\dfrac{1+y}{\sqrt{1+1}}~~\Longleftrightarrow$

$\Longleftrightarrow~~\dfrac{\sqrt{2}}{2}=\dfrac{1+y}{\sqrt{2}}~~\Longleftrightarrow$

$\Longleftrightarrow~~2(1+y)=\sqrt{2}\sqrt{2}~~\Longleftrightarrow$

$\Longleftrightarrow~~2+2y=2~~\Longleftrightarrow$

$\Longleftrightarrow~~2y=2-2~~\Longleftrightarrow$

$\Longleftrightarrow~~2y=0~~\Longleftrightarrow y=0$

Resposta: y = 0 e, portanto, P(1, 0).