Solve the equation

Question:

Let $\theta_{1}$ be the angle between two lines $2 x+3 y+c_{1}=0$ and $-x+5 y+c_{2}=0$, and $\theta_{2}$ be the angle between two lines $2 x+3 y+c_{1}=0$ and $-x+5 y+c_{3}=0$, 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-2 : $\theta_{1}=\theta_{2}$ for all $\mathrm{c}_{2}$ and $\mathrm{c}_{3}$.

  1. Statement-1 is true and Statement - 2 is true, Statement-2 is not a correct explanation for Statement-1.

  2. Statement-1 is false and Statement-2 is true.

  3. Statement-1 is true and Statement-2 is false.

  4. Statement-1 is true and Statement - 2 is true, Statement-2 is a correct explanation for Statement-1. 


Correct Option: , 4

Solution:


$\tan \theta_{1}=1=\tan \theta_{2}$

$\Rightarrow \theta_{1}=\theta_{2}$

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