(a) Show that the line $y = m x + c$ passes through the point $( 1,1 )$ if $c = 1 - m$. (b) Let $L$ be a line with gradient $m > 0$, which passes through ( 1,1 ). Find the equation of the line $L ^ { \prime }$ which is perpendicular to $L$, and which passes through the point $( 1 , a )$, given $a \neq 1$. (c) Find the area of the triangle which has ( 1,1 ) and ( $1 , a$ ) as two of its vertices and the intersection of $L$ and $L ^ { \prime }$ as the third vertex. (d) For what value of $m$ is the triangle isosceles (two sides of equal length)?
(a) Show that the line $y = m x + c$ passes through the point $( 1,1 )$ if $c = 1 - m$.
(b) Let $L$ be a line with gradient $m > 0$, which passes through ( 1,1 ). Find the equation of the line $L ^ { \prime }$ which is perpendicular to $L$, and which passes through the point $( 1 , a )$, given $a \neq 1$.
(c) Find the area of the triangle which has ( 1,1 ) and ( $1 , a$ ) as two of its vertices and the intersection of $L$ and $L ^ { \prime }$ as the third vertex.
(d) For what value of $m$ is the triangle isosceles (two sides of equal length)?