# Find the 10

Find the value of t if 2tx+5y-6=0 and 5x-4y+8=0 are perpendicular, parallel, what angle does each of the lines make with the x-axis, find the angle between the lines?

Correct result:

t1 =  2
A1 =  -38.66 °
A2 =  -128.66 °
t2 =  -3.125
B1 =  51.34 °
B2 =  -128.66 °

#### Solution:

$2tx+5y-6=0 \ \\ 5x-4y+8=0 \ \\ \ \\ n_{1}=(2t; 5) \ \\ n_{2}=(5; -4) \ \\ \ \\ \ \\ normal \ \\ n_{1}.n_{2}=0 \ \\ \ \\ \ \\ 2 \cdot \ t_{1} \cdot \ 5+5 \cdot \ (-4)=0 \ \\ \ \\ 10t_{1}=20 \ \\ \ \\ t_{1}=2$
$A_{1}=\dfrac{ 180^\circ }{ \pi } \cdot \arctan \dfrac{ 5 }{ 2 \cdot \ t_{1} } -90=\dfrac{ 180^\circ }{ \pi } \cdot \arctan \dfrac{ 5 }{ 2 \cdot \ 2 } -90=-38.66 ^\circ =-38^\circ 39'35"$
$A_{2}=\dfrac{ 180^\circ }{ \pi } \cdot \arctan \dfrac{ -4 }{ 5 } -90=-128.66 ^\circ =-128^\circ 39'35"$
$parallel \ \\ n_{1}=k \cdot \ n_{2} \ \\ k=5/(-4)=- \dfrac{ 5 }{ 4 }=-1.25 \ \\ \ \\ \ \\ 2 \cdot \ t_{2}=5/(-4) \cdot \ 5 \ \\ \ \\ 2t_{2}=-6.25 \ \\ \ \\ t_{2}=\dfrac{ -25 }{ 8 }=-3.125=- \dfrac{ 25 }{ 8 }$
$B_{1}=\dfrac{ 180^\circ }{ \pi } \cdot \arctan \dfrac{ 5 }{ 2 \cdot \ t_{2} } -90=\dfrac{ 180^\circ }{ \pi } \cdot \arctan \dfrac{ 5 }{ 2 \cdot \ (-3.125) } -90=51.34 ^\circ =51^\circ 20'25"$
$B_{2}=\dfrac{ 180^\circ }{ \pi } \cdot \arctan \dfrac{ -4 }{ 5 } -90=-128.66 ^\circ =-128^\circ 39'35"$

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