Question

if x+4y-5 =0 and 4x+ky+7=0 are two perpendicular lines then k is​

Answer 1

GIVEN :–

• Two lines are x + 4y - 5 = 0 and 4x + ky + 7 = 0.

• Lines are perpendicular.

TO FIND :–

• Value of 'k' = ?

SOLUTION :–

• Let's find the Slope of the equation x + 4y - 5 = 0 –

$\\ \implies { \bold{ M_{1} = \dfrac{ - Coefficient \: \: of \: \: x}{Coefficient \: \: of \: \: y} }}$

$\\ \implies { \bold{ M_{1} = \dfrac{ - 1}{4} \: \: \: - - - eq(1)}}$

• Slope of the equation 4x + ky + 7 = 0 –

$\\ \implies { \bold{ M_{2} = \dfrac{ - Coefficient \: \: of \: \: x}{Coefficient \: \: of \: \: y} }}$

$\\ \implies { \bold{ M_{2} = \dfrac{ - 4}{k} \: \: \: - - - eq(2)}}$

• Condition for perpendicular lines –

$\\ \implies { \bold{ M_{1}.M_{2} = - 1}}$

• Put the values –

$\\ \implies { \bold{ \left( \dfrac{ - 1}{4} \right).\left( \dfrac{ - 4}{k} \right) = - 1}}$

$\\ \implies { \bold{\left( \dfrac{ 4}{4k} \right) = - 1}}$

$\\ \implies { \bold{\left( \dfrac{ 1}{k} \right) = - 1}}$

$\\ \implies { \bold{1 = -k}}$

$\\ \: \: \dashrightarrow \: \: \large{ \boxed{ \bold{k = - 1}}}$

Answer 2

${\red{\underline{\underline{\bold{Given:-}}}}}$

• x+4y-5 = 0 and 4x +ky +7 = 0 are two perpendicular lines

${\blue{\underline{\underline{\bold{To\:Find:-}}}}}$

• The value of k

${\green{\underline{\underline{\bold{Solution:-}}}}}$

$\tt\pink{Concept\:used}$

Slope of the equation

Slope of the general equation of the line ax+by+c = 0 is

➪ $\bf m = \dfrac{ - Coefficient \: of \: x}{Coefficient \:of \: y }$

Therefore:-

$\bf \: Slope \: of \: the \: line \: x + 4y - 5 = 0 \: is \dfrac{ - 1}{4}$

$\bf \: Slope \: of \: the \: line \: 4x + ky + 7= 0 \: = \dfrac{ - 4}{k}$

Now we know that if two slopes are perpendicular the product of their slopes is -1

➪ $\bf \bigg( \dfrac{ - 1}{\:4} \bigg). \bigg(\dfrac{ - 4}{\:k} \bigg) = - 1$

➪ $\bf \bigg( \dfrac{ 1}{k} \bigg) = - 1$

➪ $\bf \: k = \dfrac{\:1}{ - 1}$

$\boxed{k=-1}$