Find the distance between parallel lines (i) 15 x+8 y-34=0 and 15 x+8 y+31=0 (ii) l(x+y)+p=0 and

Find the distance between parallel lines (i) 15 x+8 y-34=0...

Key Concepts

Standard Form Equation

A linear equation expressed in the form Ax + By + C = 0 is called the standard form. This format is particularly useful because it clearly displays the coefficients which can be used to identify the slope, intercepts, and properties of the line, serving as a basis for many geometric and algebraic analyses.

Parallel Lines

Two lines in the plane are considered parallel if they have the same slope, meaning they never intersect. In terms of equations, parallel lines have proportional coefficients for x and y when written in standard form such that the ratio A/B remains constant, which is a key criterion in establishing their parallelism.

Distance Between Parallel Lines

The distance between two parallel lines, both expressed in standard form with identical coefficients for x and y, can be calculated using the formula d = |C2 - C1| / ?(A² + B²). This formula leverages the fact that the shortest distance between two parallel lines is perpendicular to both, reflecting the constant separation throughout their lengths.

Transcript

00:01 I have to find out the distance between the parallel lines 15x plus 8y minus 34 equal to 0 and 15x plus 8y plus 31 equal to 0.

00:16 So it's already given to us that these are these two lines are parallel and we just have to find out the distance over here.

00:23 So now we know that the distance between two parallel lines that is of the form a x plus b y plus c1 equal to zero and a x plus b y plus c2 equal to zero is always given by b is equal to modules of c1 minus c2 upon under root of a square plus b square right so if you just consider the first equation over here for the first line let's say this is one and this is two so for equation one over here the above equation is of the form a x plus b y plus c1 where a is equal to fifth b is equal to 8 and c1 is equal to minus 34 and for equation 2 if we look then a is equal to 15 b is equal to 8 and c2 is equal to 31 right we know the formula if you just plug in the values over here in this formula you will get the required distance so d over here will be equal to modulus of c1 minus c2 that means minus 34 minus 31 upon model or sorry under root of a square plus b square that means 15 square plus a square so this will be equal to minus 65 model is so minus 65 upon under root of 225 plus 64 that is equal to 289 so this will be equal to 65 since this is model is that means i'll consider only the positives over here so minus minus modulus of minus 65 is equal to 65 and root of 289 is equal to 17.

02:07 Therefore, the required distance over here is 65 upon 17 units.

02:14 So this will be your answer.

02:16 Moving on to the second equation now, the two equations over here this time are l times x plus y plus p equal to 0.

02:26 And the other equation is l times x plus y minus r equal to zero...

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