Line p has equation y=x+3, and line q has equation y=x-1. a....

Line $p$ has equation $y=x+3,$ and line $q$ has equation $y=x-1$. a. Find the equation of a line $r$ that is perpendicular to $p$ and $q .$ b. Find the coordinates of the intersection of $p$ and $r$ and the coordinates of the intersection of $q$ and $r$. c. Find the distance between lines $p$ and $q$.

Line $p$ has equation $y=x+3,$ and line $q$ has equation $y=x-1$.
a. Find the equation of a line $r$ that is perpendicular to $p$ and $q .$
b. Find the coordinates of the intersection of $p$ and $r$ and the coordinates of the intersection of $q$ and $r$.
c. Find the distance between lines $p$ and $q$.

Key Concepts

Intersection of Lines

Finding the intersection of lines involves solving a system of linear equations simultaneously. The solution corresponds to the point where the two lines meet on the coordinate plane, which is essential for determining relationships between lines.

Distance Between Parallel Lines

This concept refers to the shortest distance between two parallel lines, which can be calculated by using the formula that involves the absolute difference of the y-intercepts divided by the square root of 1 plus the square of the slope, providing a measure of separation between the lines.

Slope of a Line

This concept involves understanding that the slope (m) in the line equation y = mx + b indicates the rate at which y changes with respect to x, and it is crucial for determining the angle and direction of the line in the coordinate plane.

Perpendicular Lines

In coordinate geometry, when two lines are perpendicular, the product of their slopes is -1. This means if one line has a slope m, the other must have a slope of -1/m, a property used to find equations of lines perpendicular to given lines.

Transcript

00:01 Problem number 52 we have two lines that is one line is p whose equation is y equal to x plus 3 another line is q whose equation is x minus 1 if we compare with y equal to mx plus c so slope of this line let us say m1 becomes 1 and slope of this line let us say m2 becomes also 1 so here m1 and m2 are m1 equal to m2 so these two lines are parallel parallel lines okay so let us draw some parallel lines to get into the concept parallel lines now we need to find the equation of line which is perpendicular to both these lines so we need to find the equation of line which is perpendicular to both these lines so we can see see that there can be infinite number of lines infinite number of lines which can be perpendicular to these two lines because these two lines are parallel so this is infinite number of lines and cannot be determined by any equation just we can write as there will be infinity infinite number of equation number of equation of lines which are perpendicular to both these lines and it depends on any of the point given through which this line passes it means if we would have been given that find the equation of line which is perpendicular to these two lines and passing through this so we could have calculated easily we can also calculate this easily but we need to get any point on this line or any point on this line so if we go in general way so let us assume if this line be y equal to x plus 3 if this line is y equal to x plus 3 if this line is y equal to x plus 3 x plus 3 let us assume any point on this line be any point on this line be here i'm talking about this point be 0 .3 because when plugging in x equal to 0 we could have find y equal to 3 so 0 .3 now according to this we will go and we will go for the solution if we need to otherwise there will be infinite number of solutions so we have slope of perpendicular.

04:15 What will be slope of perpendicular negative minus 1 over slope of this line 1 that is minus 1 and passing through the point 0 comma 3.

04:24 So we will be having equation y minus 3 equal to minus 1 x minus 0 that is y equal to minus x plus 3.

04:36 This should be one of the...

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