Question

Find the value of $k$ for which the lines $kx-5y+4=0$ and $5x+2y+5=0$ are perpendicular to each other.

Answer 1

Given

Line $1:5x-5y+4=0$

Line $2:4x+ky+5=0$

General equation for line

is $y=mx+c\to \left(i\right)$

converting both line equations

as (i),

for line 1

$5x-5y+4=0$

$5y=5x+4$

$y=x+\frac{4}{5}\to \left(ii\right)$

Comparing eq${}^n$ (ii) with (i)

$m_1=\left(1\right)$

Now for line 2

$4x+ky+5=0$

$\therefore -ky=4x+5$

$\therefore y=\frac{-4}{k}x-5\to \left(iii\right)$

Comparing eq${}^n$ (iii) with (i)

$m_2=(-4/k)$

For any two lines which are

perpendicular to each other

$m_1.m_2=(-1)\to \left(iv\right)$

$\therefore$ Substituting the values of

$m_1$ & $m_2$ in eq${}^n$ (iv)

$\left(1\right).\left(\frac{-4}{k}\right)=(-1)$

$\therefore k=4$