Question

1- Qual é a distância entre o ponto P(2, 3) e a reta de equação 4x + 3y - 12 = 0.
( ) 2
( ) 3
( ) 1
( ) 4

2- Escreva a equação da reta que passa pelo ponto P(3, -3) e é paralela à reta 2x – 3y -6 = 0
( ) 2x – y + 9 = 0
( ) 2x – 3y – 15 = 0
( ) 3x + 2y – 15 = 0
( ) x – 2y + 9 = 0

3- Determine a equação da reta que passa pelos pontos A(-3, 2) e B(5, -4)
15 pontos
( ) 4x + 3y + 1= 0
( ) 3x + 4y + 1= 0
( ) x + y + 3 = 0
( ) x + y – 4 = 0

4) As retas de equações r: x - 2y + 1 = 0 e s: -x - 3y - 1 = 0 são
( ) perpendiculares
( ) concorrentes
( ) paralelas

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\rm Dist\hat ancia~do~ponto~\grave a~reta}\\\sf Dada~uma~reta\\\sf r:ax+by+c=0~e~um~ponto~P(x_P,y_P)\\\sf a~dist\hat ancia~de~P~\grave a~r~\acute e\\\sf d_{P,r}=\dfrac{| ax_P+by_P+c|}{\sqrt{a^2+b^2}}\end{array}}$

$\large\boxed{\begin{array}{l}\rm 1)~Qual~\acute e~a~dist\hat ancia~entre~o~ponto~P(2,3)\\\rm e~a~reta~de~equac_{\!\!,}\tilde ao~4x+3y-12=0?\\\tt a)~2\\\tt b)~3\\\tt c)~1\\\tt d)~4\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm soluc_{\!\!,}\tilde ao\!:}\\\sf r:4x+3y-12=0\implies a=4~~b=3~~c=-12\\\sf P(2,3)\implies x_P=2~~y_P=3\\\sf d_{P,r}=\dfrac{|ax_P+by_P|}{\sqrt{a^2+b^2}}\\\\\sf d_{P,r}=\dfrac{|4\cdot2+3\cdot3-12|}{\sqrt{4^2+3^2}}\\\\\sf D_{P,r}=\dfrac{5}{\sqrt{25}}=\dfrac{5}{5}=1\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~c}}}}\end{array}}$

$\large\boxed{\begin{array}{l}\rm 2)~Escreva~a~equac_{\!\!,}\tilde ao~da~reta\\\rm que~passa~pelo~ponto~P(3,-3)\\\rm e~\acute e~paralela~\acute a~reta~2x-3y-6=0\\\tt a)~2x-y+9=0\\\tt b)~2x-3y-15=0\\\tt c)~3x+2y-15=0\\\tt d)~x-2y+9=0\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm soluc_{\!\!,}\tilde ao\!:}\\\sf Seja~r~a~reta~de~equac_{\!\!,}\tilde ao~ax+by+c=0\\\sf que~\acute e~paralela~a~reta~s:2x-3y-6=0\\\sf se~r\parallel s~en\tilde tao~m_r=m_s\\\sf s:2x-3y-6=0\implies y=\dfrac{2}{3}x-2\\\sf m_s=\dfrac{2}{3}\implies m_r=\dfrac{2}{3}\\\sf r~passa~pelo~ponto~P(3,-3)~logo\\\sf y=y_0+m_r(x-x_0)\\\sf y=-3+\dfrac{2}{3}(x-3)\\\sf y=-3+\dfrac{2}{3}x-2\\\sf y=\dfrac{2}{3}x-5\cdot(3)\\\sf 3y=2x-15\implies r:2x-3y-15=0\end{array}}$

$\Huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~b}}}}$

$\large\boxed{\begin{array}{l}\rm 3)~Determine~a~equac_{\!\!,}\tilde ao~da~reta\\\rm que~passa~pelos~pontos~A(-3,2)~e~B(5,-4).\\\tt a)~4x+3y+1=0\\\tt b)~3x+4y+1=0\\\tt c)~x+y+3=0\\\tt d)~x+y-4=0\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm soluc_{\!\!,}\tilde ao\!:}\\\begin{vmatrix}\sf-3&\sf2&\sf1\\\sf5&\sf-4&\sf1\\\sf x&\sf y&\sf1\end{vmatrix}=0\\\sf -3(-4-y)-2(5-x)+1\cdot(5y+4x)=0\\\sf12+3y-10+2x+5y+4x=0\\\sf6x+8y+2=0\div(2)\\\sf 3x+4y+1=0\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~b}}}}\end{array}}$

$\large\boxed{\begin{array}{l}\sf Considere~ duas~retas\\\sf\ell_1:m_1x+n_1\\\sf\ell_2:m_2x+n_2\\\underline{\rm ent\tilde ao\!:}\\\sf m_1=m_2~e~n_1\ne n_2\implies\,\ell_1~e~\ell_2~paralelas\\\sf m_1=m_2~e~n_1=n_2\implies\,\ell_1~e~\ell_2~coincidem\\\sf m_1\ne m_2\implies\,\ell_1~e~\ell_2~concorrentes\\\sf m_1\cdot m_2=-1\implies\,\ell_1~e~\ell_2~perpendiculares\end{array}}$

$\large\boxed{\begin{array}{l}\rm 4)As~retas~de~equac_{\!\!,}\tilde oes\\\sf r:x-2y+1=0~e~r=-x-3y-1=0\\\rm s\tilde ao:\\\tt a)~perpendiculares\\\tt b)~concorrentes\\\tt c)~~ paralelas\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm soluc_{\!\!,}\tilde ao:}\\\sf r:x-2y+1=0\implies y=\dfrac{1}{2}x+\dfrac{1}{2}\\\sf m_r=\dfrac{1}{2}\\\sf s: -x-3y-1=0\implies y=-\dfrac{1}{3}x-\dfrac{1}{3}\\\sf m_s=-\dfrac{1}{3}\\\sf como~m_r\ne m_s\implies r~e~s~s\tilde ao~concorrentes\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~b}}}}\end{array}}$