If two lines perpendicular to each other always satisfy the condition -1-2-2- where -1 and -2 are the angles made by the lines with positive direction of X - axis and M 1- M 2 be the respective slope- then - M 1-2 M 2- is equal to where M 1- M 2

1M_1+1M_2=2 ⇒ M_1+M_2= 2 (∵ M_1M_2= 1) ⇒ M_1 1M_1= 2 ⇒ M_1^2+2M_1 1=0 ⇒ M_1= 1±√(2) ∴ M_1=√(2) 1, M_2= √(2) 1 (∵ M_1 M_2) ∴ |M_1+√(2) M_2|=3 ...

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Byju's Answer

Standard XII

Mathematics

Perpendicular Form of a Straight Line

If two lines ...

Question

If two lines perpendicular to each other always satisfy the condition $cot θ_{1} + cot θ_{2} = 2,$ where $θ_{1}$ and $θ_{2}$ are the angles made by the lines with positive direction of $X -$ axis and $M_{1}, M_{2}$ be the respective slope, then $| M_{1} + \sqrt{2} M_{2} |$ is equal to
(where $M_{1} > M_{2}$ )

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Solution

$\frac{1}{M_{1}} + \frac{1}{M_{2}} = 2 \Rightarrow M_{1} + M_{2} = - 2 (∵ M_{1} M_{2} = - 1) \Rightarrow M_{1} - \frac{1}{M_{1}} = - 2 \Rightarrow M_{1}^{2} + 2 M_{1} - 1 = 0 \Rightarrow M_{1} = - 1 \pm \sqrt{2} ∴ M_{1} = \sqrt{2} - 1, M_{2} = - \sqrt{2} - 1 (∵ M_{1} > M_{2})$

$∴ | M_{1} + \sqrt{2} M_{2} | = 3$

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Q. If two lines perpendicular to each other always satisfy the condition

cot θ_{1} + cot θ_{2} = 2,

where

θ_{1}

and

θ_{2}

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X -

axis and

M_{1},

M_{2}

be the respective slopes, then

∣ ∣ M_{1} + \sqrt{2} M_{2} ∣ ∣

is equal to
(where

M_{1} > M_{2}

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Let the angle made by the line OA with respect to positive x-axis be $θ_{1}$ and the angle made by OB be $θ_{2}$ . Find the value of $θ_{1} + θ_{2} (0 \leq | θ_{1} |, | θ_{2} | \leq 180^{\circ})$ . Angle measured in clockwise direction is negative and angle measured in anti-clockwise direction is positive.