State whether the statement is true/false.

The lines $2 x + 3 y = 19$ and $2 x + 3 y + 7 = 0$ are not equidistant from the line $2 x + 3 y = 6$ .

True

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False

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Solution

The correct option is B False
Let $d_{1}$ be the distance between the lines $2 x + 3 y = 19$ and $2 x + 3 y = 6$
while $d_{2}$ be the distance between the lines $2 x + 3 y + 7 = 0$ and $2 x + 3 y = 6$
$∴ d_{1} = ∣ ∣ ∣ ∣ \frac{- 19 - (- 6)}{\sqrt{2^{2} + 3^{2}}} ∣ ∣ ∣ ∣$ and $d_{2} = ∣ ∣ ∣ ∣ \frac{7 - (- 6)}{\sqrt{2^{2} + 3^{2}}} ∣ ∣ ∣ ∣$
$\Rightarrow d_{1} = ∣ ∣ ∣ \frac{- 13}{\sqrt{13}} ∣ ∣ ∣ = \sqrt{13}$ and $d_{2} = ∣ ∣ ∣ \frac{13}{\sqrt{13}} ∣ ∣ ∣ = \sqrt{13}$
Hence, the lines $2 x + 3 y = 19$ and $2 x + 3 y + 7 = 0$ are equidistant from the line $2 x + 3 y = 6$
Hence the given statement is false.

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