Let _1 be the angle between two lines 2x+3y+c_1=0 and -x+5y+c_2=0 and _2 be the angle betw…

Question

Let $	heta_1$ be the angle between two lines $2x+3y+c_1=0$ and $-x+5y+c_2=0$ and $	heta_2$ be the angle between two lines $2x+3y+c_1=0$ and $-x+5y+c_3=0$, where $c_1,c_2,c_3$ are any real numbers:

Statement-1: If $c_2$ and $c_3$ are proportional, then $	heta_1=	heta_2$.

Statement-2: $	heta_1=	heta_2$ for all $c_2$ and $c_3$

Accepted Answer

The lines $-x+5y+c_2=0$ and $-x+5y+c_3=0$ are always parallel to each other (same slope, regardless of $c_2,c_3$), so each makes the same angle with the fixed line $2x+3y+c_1=0$, i.e. $	heta_1=	heta_2$ always. This proves Statement-2 is true, and it correctly explains why Statement-1 (a weaker, specific case) holds.

Solutions · 1