Question

# Find the equation of a line which passes through (1, 2) and is perpendicular to the line with an equation y = x – 1.

## The correct answer is: x + y = 3 is the equation of the given line

### Hint:-

1. The slope of a line can be defined as the change in y coordinates of any 2 points on that line corresponding to the change in the x coordinates of those 2 points. This is generally referred to as the rise to run ratio of the given line i.e. how much did the y-coordinates rise vis-a-vis how long a distance was covered by the x-coordinates. Slope = m = rise / run = y2-y1 / x2-x1

2. Parallel lines have equal slopes.

3. Equation of a line in slope point form is-

(y-y1) = m (x - x1)

Step-by-step solution:-

Let l be the line for which slope is to be found.

Comparing the equation y = x – 1 with standard form of a straight line i.e. y = mx + c, we get-

m = 1 ….................................................................................................................... (Equation i)

Now, line l is perpendicular to this line (y = x – 1) .............................................. (Given)

and we know that- Slopes of perpendicular lines are negative reciprocals of each other.

∴ Slope of line l = -1 / slope of line (y = x - 1)

∴ Slope of line l = -1 / 1 = -1 ....................... (From Equation i) ............................ (Equation ii)

We are given that line l passes through the point (1,2) and we know its slope.

i.e. x1 = 1 & y1 = 2 & m = -1

We can use slope point form of an equation to find the equation of line l-

(y - y1) = m (x - x1)

∴ (y - 2) = -1 (x - 1)

∴ y - 2 = -x + 1

∴ x + y = 1 + 2

∴ x + y = 3

Final Answer:-

∴ x + y = 3 is the equation of the given line.

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