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Question Number 25971 by Tinkutara last updated on 17/Dec/17
In finding the equations of the bisectors of the angles between two lines a_1 x+b_1 y+c_1 =0 and a_2 x+b_2 y+c_2 =0, why we observe a_1 a_2 +b_1 b_2 >0 or <0 for obtuse and acute angle bisectors?
$${In}\:{finding}\:{the}\:{equations}\:{of}\:{the} \\ $$$${bisectors}\:{of}\:{the}\:{angles}\:{between}\:{two} \\ $$$${lines}\:{a}_{\mathrm{1}} {x}+{b}_{\mathrm{1}} {y}+{c}_{\mathrm{1}} =\mathrm{0}\:{and}\:{a}_{\mathrm{2}} {x}+{b}_{\mathrm{2}} {y}+{c}_{\mathrm{2}} =\mathrm{0}, \\ $$$${why}\:{we}\:{observe}\:{a}_{\mathrm{1}} {a}_{\mathrm{2}} +{b}_{\mathrm{1}} {b}_{\mathrm{2}} >\mathrm{0}\:{or}\:<\mathrm{0} \\ $$$${for}\:{obtuse}\:{and}\:{acute}\:{angle}\:{bisectors}? \\ $$
Commented by Tinkutara last updated on 17/Dec/17
Commented by Tinkutara last updated on 17/Dec/17
For example here, why we are calculating a_1 a_2 +b_1 b_2 ? Why not a_1 b_1 +a_2 b_2 or something else?
$${For}\:{example}\:{here},\:{why}\:{we}\:{are} \\ $$$${calculating}\:{a}_{\mathrm{1}} {a}_{\mathrm{2}} +{b}_{\mathrm{1}} {b}_{\mathrm{2}} ?\:{Why}\:{not} \\ $$$${a}_{\mathrm{1}} {b}_{\mathrm{1}} +{a}_{\mathrm{2}} {b}_{\mathrm{2}} \:{or}\:{something}\:{else}? \\ $$
Answered by ajfour last updated on 17/Dec/17
to check if ∣θ_2 −θ_1 ∣ is obtuse or acute we look for sign of cos (θ_2 −θ_1 ). ∣θ_2 −θ_1 ∣ is acute only if cos (θ_2 −θ_1 ) > 0 , and cos θ_1 =((−b_1 )/( (√(a_1 ^2 +b_1 ^2 )))) ,sin θ_1 =(a_1 /( (√(a_1 ^2 +b_1 ^2 )))) cos θ_2 =((−b_2 )/( (√(a_2 ^2 +b_2 ^2 )))) , sin θ_2 =(a_2 /( (√(a_2 ^2 +b_2 ^2 )))) so cos (θ_2 −θ_1 ) > 0 ⇒ cos θ_2 cos θ_1 +sin θ_2 sin θ_1 > 0 or b_1 b_2 +a_1 a_2 > 0 .
$${to}\:{check}\:{if}\:\mid\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \mid\:{is}\:{obtuse}\:{or} \\ $$$${acute}\:\:{we}\:{look}\:{for}\:{sign}\:{of}\: \\ $$$$\mathrm{cos}\:\left(\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \right). \\ $$$$\mid\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \mid\:{is}\:{acute}\:{only}\:{if} \\ $$$$\mathrm{cos}\:\left(\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \right)\:>\:\mathrm{0}\:,\:\:{and} \\ $$$$\mathrm{cos}\:\theta_{\mathrm{1}} =\frac{−{b}_{\mathrm{1}} }{\:\sqrt{{a}_{\mathrm{1}} ^{\mathrm{2}} +{b}_{\mathrm{1}} ^{\mathrm{2}} }}\:\:,\mathrm{sin}\:\theta_{\mathrm{1}} =\frac{{a}_{\mathrm{1}} }{\:\sqrt{{a}_{\mathrm{1}} ^{\mathrm{2}} +{b}_{\mathrm{1}} ^{\mathrm{2}} }} \\ $$$$\:\mathrm{cos}\:\theta_{\mathrm{2}} =\frac{−{b}_{\mathrm{2}} }{\:\sqrt{{a}_{\mathrm{2}} ^{\mathrm{2}} +{b}_{\mathrm{2}} ^{\mathrm{2}} }}\:,\:\mathrm{sin}\:\theta_{\mathrm{2}} =\frac{{a}_{\mathrm{2}} }{\:\sqrt{{a}_{\mathrm{2}} ^{\mathrm{2}} +{b}_{\mathrm{2}} ^{\mathrm{2}} }}\:\:\:\: \\ $$$${so}\:\:\mathrm{cos}\:\left(\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \right)\:>\:\mathrm{0} \\ $$$$\Rightarrow\:\:\mathrm{cos}\:\theta_{\mathrm{2}} \mathrm{cos}\:\theta_{\mathrm{1}} +\mathrm{sin}\:\theta_{\mathrm{2}} \mathrm{sin}\:\theta_{\mathrm{1}} \:>\:\mathrm{0} \\ $$$${or}\:\:\:\:{b}_{\mathrm{1}} {b}_{\mathrm{2}} +{a}_{\mathrm{1}} {a}_{\mathrm{2}} \:>\:\mathrm{0}\:. \\ $$$$\:\:\:\:\:\: \\ $$
Commented by Tinkutara last updated on 17/Dec/17
What are θ_1 and θ_2 ?
$${What}\:{are}\:\theta_{\mathrm{1}} \:{and}\:\theta_{\mathrm{2}} ? \\ $$
Commented by ajfour last updated on 17/Dec/17
Angles of lines with +ve x axis, equations of whose bisectors we seek.
$${Angles}\:\:{of}\:{lines}\:{with}\:+{ve}\:{x}\:{axis}, \\ $$$${equations}\:{of}\:{whose}\:{bisectors}\:{we} \\ $$$${seek}. \\ $$
Commented by Tinkutara last updated on 17/Dec/17
Sir can you prove exactly Step 3?

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