Equation of line in Straight lines

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If the line $2 x - y + 3 = 0$ is at distances $\frac{1}{\sqrt{5}} and$ from the lines $4 x - 2 y + α = 0 and 6 x - 3 y + β = 0$ respectively the sum of all possible values of a and $β,$ , is

Moderate

Solution

It is given that
$\begin{array}{l} |\frac{3 - α}{2}| = \frac{1}{\sqrt{5}} and \frac{|3 - \frac{β}{3}|}{\sqrt{4 + 1}} = \frac{2}{\sqrt{5}} \\ \Rightarrow |3 - \frac{α}{2}| = 1 and |3 - \frac{β}{3}| = 2 \\ \Rightarrow | 6 - α | = 2 and | 9 - β | = 6 \end{array}$
$\begin{array}{l} \Rightarrow 6 - α = \pm 2 and 9 - β = \pm 6 \\ \Rightarrow α = 4, 8 and β = 3, 15 \end{array}$
Hence, the sum of all possible values of $α and β$ is 4 + 8 + 3 + 15 = 30 .

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Equation of line in Straight lines

Remember concepts with our Masterclasses.

If the line 2x−y+3=0 is at distances 15 and from the lines 4x−2y+α=0 and 6x−3y+β=0 respectively the sum of all possible values of a and β,, is

It is given that3−α2=15 and 3−β34+1=25⇒3−α2=1 and 3−β3=2⇒|6−α|=2 and |9−β|=6⇒ 6−α=±2 and 9−β=±6⇒ α=4,8 and β=3,15Hence, the sum of all possible values of α and β is 4 + 8 + 3 + 15 = 30 .

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free study material

If the line $2 x - y + 3 = 0$ is at distances $\frac{1}{\sqrt{5}} and$ from the lines $4 x - 2 y + α = 0 and 6 x - 3 y + β = 0$ respectively the sum of all possible values of a and $β,$ , is