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  4. Let θ1 be the angle between two lines 2x + 3y + C1 = 0 and - x + 5y + c2 = 0 and θ2 be the angle between two fines 2x + 3y + cx = 0 and -x + 5y+ c3 = 0, where c1 c2, c3 are any real numbers: Statement-1: If c2 and c3 are proportional, then θ1 = θ2. Statement-1: θ1=θ2 for all c2 and c3

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Q. Let $\theta_{1}$ be the angle between two lines $2x + 3y + C_1 = 0$ and $- x + 5y + c_2 = 0$ and $\theta_{2}$ be the angle between two fines $2x + 3y + cx = 0$ 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 $\theta_{1} = \theta_{2}$.
Statement-1: $\theta_{1}=$$\theta_{2}$ for all $c_2$ and $c_3$

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Solution:

Two lines $- x + 5y+c_2=0$ and$- x + 5y+c_3 = 0$ are parallel to each other. Hence statement-1 is true, statement-2 is true and statement-2 is the correct explanation of statement-1.