Show that the equation of the parallel line midway between the paralle

1.	Show that the equation of the parallel line midway between the parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is as + by + \(\frac{c_1+c_2}{2}\) = 0.
Answer» The two given lines are ax + by + c₁ = 0 and ax + by + c₂ = 0. Any line parallel to these two lines and midway between them is ax + by + c = 0 ...(i) Putting x = 0, y = \(-\frac{c}{b}\) is a point on line (i) It is equidistant from the given lines but in opposite directions, so \(\frac{\big\|a\times0+b\times-\frac{c}{b}+c_1\big\|}{\sqrt{a^2+b^2}}\) = \(-\frac{\big\|a\times0+b\times-\frac{c}{b}+c_2\big\|}{\sqrt{a^2+b^2}}\) ⇒ – c + c₁ = c – c₂⇒ c = \(\frac{c_1+c_2}{2}\) ∴ Required equation is ax + by + \(\frac{c_1+c_2}{2}\) = 0.

Show that the equation of the parallel line midway between the parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is as + by + \(\frac{c_1+c_2}{2}\) = 0.

Answer»

The two given lines are ax + by + c₁ = 0 and ax + by + c₂ = 0.

Any line parallel to these two lines and midway between them is

ax + by + c = 0 ...(i)

Putting x = 0, y = \(-\frac{c}{b}\) is a point on line (i)

It is equidistant from the given lines but in opposite directions, so

\(\frac{\big|a\times0+b\times-\frac{c}{b}+c_1\big|}{\sqrt{a^2+b^2}}\) = \(-\frac{\big|a\times0+b\times-\frac{c}{b}+c_2\big|}{\sqrt{a^2+b^2}}\) ⇒ – c + c₁ = c – c₂⇒ c = \(\frac{c_1+c_2}{2}\)

∴ Required equation is ax + by + \(\frac{c_1+c_2}{2}\) = 0.

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