5.Does line 3x + 4y = 7 pass through Origin?
O yes
O no
O not sure
1S
O non of all
Answers
Among all the options the answer is NOT SURE.........
Step-by-step explanation:
Given
→ Two parallel straight line are given :-
\begin{lgathered}\sf{\implies \: 3x + 4y = 12 \: and \: 6x + 8y + 3 = 0}\\\end{lgathered}
⟹3x+4y=12and6x+8y+3=0
→ A line passing through origin intersects these two lines at P and Q point.
\begin{lgathered}\large{\underline{\underline{\sf{To \: Find}}}}\\\end{lgathered}
ToFind
→ The ratio of OP and OQ .
\begin{lgathered}\large{\underline{\underline{\sf{Solution}}}}\\\end{lgathered}
Solution
Diagram refers to the attachment .
Let's assume that the ratio is k : 1.
So first of all we will solve the first equation by using the equation of line passing through the origin .
\begin{lgathered}\sf{\implies \: 3x+4y=12\: and\: y= mx}\\\end{lgathered}
⟹3x+4y=12andy=mx
Putting the value of y in first equation .
\begin{lgathered}\sf{\implies 3x + 4(mx) = 12 }\\\end{lgathered}
⟹3x+4(mx)=12
\begin{lgathered}\sf{\implies 3x + 4mx = 12 }\\\end{lgathered}
⟹3x+4mx=12
\begin{lgathered}\sf{\implies x( 3 + 4m) = 12 }\\\end{lgathered}
⟹x(3+4m)=12
\begin{lgathered}{\underline{\underline{\sf{\implies x = \frac{12}{3+4m} }}}}\\\end{lgathered}
⟹x=
3+4m
12
Now putting the value of x in equation first so that we can get the value of y also.
\begin{lgathered}\sf{\implies 3( \frac{12}{3 + 4m} ) + 4y = 12 }\\\end{lgathered}
⟹3(
3+4m
12
)+4y=12
Taking 4 common from both sides
\begin{lgathered}\sf{\implies 4 [ ( \frac{9}{3 + 4m} + y ] = 4[ 3 ] }\\\end{lgathered}
⟹4[(
3+4m
9
+y]=4[3]
\begin{lgathered}\sf{\implies \frac{9 + 3y + 4ym }{ 3+4m} = 3 }\\\end{lgathered}
⟹
3+4m
9+3y+4ym
=3
\begin{lgathered}\sf{\implies 9 + 3y + 4ym = 9 + 12m }\\\end{lgathered}
⟹9+3y+4ym=9+12m
\begin{lgathered}\sf{\implies y(3 + 4m) = 9-9+12m }\\\end{lgathered}
⟹y(3+4m)=9−9+12m
\begin{lgathered}{\underline{\underline{\sf{\implies y = \frac{12m}{3+4m} }}}}\\\end{lgathered}
⟹y=
3+4m
12m
\begin{lgathered}{\boxed{\sf{Coordinates\: of \: P( \frac{12}{3+4m} , \frac{12m}{3+4m}}}}\\\end{lgathered}
CoordinatesofP(
3+4m
12
,
3+4m
12m
Now we will solve equation second by the equation of line passing through origin .
\begin{lgathered}\sf{\implies 6x + 8y = -3 \: and \: y = mx }\\\end{lgathered}
⟹6x+8y=−3andy=mx
Putting the value of y in equation second.
\begin{lgathered}\sf{\implies 6x + 8(mx) = -3 }\\\end{lgathered}
⟹6x+8(mx)=−3
\begin{lgathered}\sf{\implies 6x + 8mx = -3 }\\\end{lgathered}
⟹6x+8mx=−3
\begin{lgathered}\sf{\implies x(6+8m) = -3 }\\\end{lgathered}
⟹x(6+8m)=−3
\begin{lgathered}{\underline{\underline{\sf{\implies x = \frac{-3}{6 + 8m} }}}}\\\end{lgathered}
⟹x=
6+8m
−3
Now putting the value of x in equation second so that we can get the value of y also.
\begin{lgathered}\sf{\implies 6[ \frac{-3}{6 + 8m} ] + 8y = -3 }\\\end{lgathered}
⟹6[
6+8m
−3
]+8y=−3
\begin{lgathered}\sf{\implies [ \frac{-18}{6+8m} ] + 8y = -3 }\\\end{lgathered}
⟹[
6+8m
−18
]+8y=−3
\begin{lgathered}\sf{\implies \frac{- 18 + 48y + 64my}{6+8m} = -3 }\\\end{lgathered}
⟹
6+8m
−18+48y+64my
=−3
\begin{lgathered}\sf{\implies -18 + 48y + 64my = -18 - 24m }\\\end{lgathered}
⟹−18+48y+64my=−18−24m
\begin{lgathered}\sf{\implies y( 48 + 64m) = -24m }\\\end{lgathered}
⟹y(48+64m)=−24m
\begin{lgathered}{\underline{\underline{\sf{\implies y = \frac{-24}{ 48 + 64 m } }}} }\\\end{lgathered}
⟹y=
48+64m
−24
Taking 8 common from RHS fraction .
\begin{lgathered}{\underline{\underline{\sf{\implies y = \frac{-3m}{6 + 8m} }}}}\\\end{lgathered}
⟹y=
6+8m
−3m
\begin{lgathered}\sf{Coordinates\: of \: Q ( \frac{-3}{6+8m} , \frac{-3m}{6+8m} }\\\end{lgathered}
CoordinatesofQ(
6+8m
−3
,
6+8m
−3m
Now we know co-ordinate of origin are O( 0,0)
Using intersection formula.
\begin{lgathered}\sf{\implies Ox = \frac{m_1 ( x_2) + m_2(x_1)}{k+1} }\\\end{lgathered}
⟹Ox=
k+1
m
1
(x
2
)+m
2
(x
1
)
\begin{lgathered}\sf{\implies Ox = \frac{k ( \frac{-3}{6+8m} + 1( \frac{12}{3+4m}}{k+1} }\\\end{lgathered}
⟹Ox=
k+1
k(
6+8m
−3
+1(
3+4m
12
→ 0 = [-3k/6+8m + 12/3+4m]/[k+1]
→ 0 = [-3k/2 + 12]
→ -12 = -3k/2
→ -24 = -3k
→ 24/3 = k
→ 8/ 1 = k/1
So Ratio is 8 : 1