**Equation of the Line Passing Through the Point :**

Here we are going to see how to find the equation of the line passing through the point, which is parallel or perpendicular to the given line.

Before going to see example problems, we must know the following things.

If two lines are parallel |
If two lines are perpendicular |

**Question 1 :**

Write the equation of the lines through the point (1,−1)

(i) parallel to x + 3y − 4 = 0

(ii) perpendicular to 3x + 4y = 6

**Solution :**

(i) Since the required line is parallel to the line x + 3y − 4 = 0 is, slopes of the required line and given line will be equal.

Slope of the line x + 3y − 4 = 0

= - Coefficient of x/coefficient of y = -1/3

Slope of the required line = -1/3

Equation of the required line :

(y - y_{1}) = m (x - x_{1})

(y - 1) = (-1/3) (x - 1)

3(y - 1) = -1(x - 1)

3y - 3 = -x + 1

x + 3y - 4 = 0

(ii) perpendicular to 3x + 4y = 6

Since the required line is perpendicular to the given line, product of their slopes will be equal to -1.

Slope of the line 3x + 4y = 6

= - Coefficient of x/coefficient of y = -3/4

Slope of the required line = 4/3

Equation of the required line :

(y - y_{1}) = m (x - x_{1})

(y - 1) = (4/3) (x - 1)

3(y - 1) = 4(x - 1)

3y - 3 = 4x - 4

4x - 3y - 4 + 3 = 0

4x - 3y - 1 = 0

**Question 2 :**

If (−4, 7) is one vertex of a rhombus and if the equation of one diagonal is 5x − y + 7 = 0, then find the equation of another diagonal.

**Solution :**

In a rhombus, both diagonals will intersect each other at right angle.

So, the required diagonal will be perpendicular to the line 5x - y + 7 = 0 and passing through the point (-4, 7).

Slope of the line = Coefficient of x/Coefficient of y

= -5/(-1) = 5

Slope of required diagonal = -1/5

Equation of other diagonal :

(y - y_{1}) = m (x - x_{1})

(y - 7) = (-1/5) (x + 4)

5(y - 7) = -1(x + 4)

5y - 35 = -x - 4

x + 5y - 35 + 4 = 0

x + 5y - 31 = 0

After having gone through the stuff given above, we hope that the students would have understood "Equation of the Line Passing Through the Point".

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