Let theta_1 be the angle between two lines 2x | vedclass

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(A) The angle $	heta$ between two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is given by $	an 	heta = \left| \frac{a_1b_2 - a_2b_1}{

Vedclass · Accepted Answer

Statement-$1$ is true,Statement-$2$ is true; Statement-$2$ is a correct explanation of Statement-$1$.

Vedclass · Answer

(A) The angle $	heta$ between two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is given by $	an 	heta = \left| \frac{a_1b_2 - a_2b_1}{a_1a_2 + b_1b_2} ight|$.For the lines $2x + 3y + c_1 = 0$ and $-x + 5y + c_2 = 0$,the angle $	heta_1$ satisfies $	an 	heta_1 = \left| \frac{(2)(5) - (3)(-1)}{(2)(-1) + (3)(5)} ight| = \left| \frac{10 + 3}{-2 + 15} ight| = \frac{13}{13} = 1$.Similarly,for the lines $2x + 3y + c_1 = 0$ and $-x + 5y + c_3 = 0$,the angle $	heta_2$ satisfies $	an 	heta_2 = \left| \frac{(2)(5) - (3)(-1)}{(2)(-1) + (3)(5)} ight| = 1$.Since $	an 	heta_1 = 	an 	heta_2 = 1$,we have $	heta_1 = 	heta_2$ for any real values of $c_2$ and $c_3$.Thus,Statement-$2$ is true.Since Statement-$2$ is true,Statement-$1$ is also true,and Statement-$2$ provides the logical basis for Statement-$1$.

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