Question

Find the equation of the line that is perpendicular to the line y = 3x –1 and passes through the point (7,4)

Answer 1

The equation of required line is y = -1/3 x + 7

Given,

Equation of line L1: y = 3x - 1

Line L passes through the point (x1, y1) = (7,4)

To Find,

The equation of line L.

Solution,

The equation of line whose slope and a point which it passes through are known is given by:

y - y1 = m (x - x1)

where m is the slope of the line and (x1, y1) is the point through which it passes.

Let our required line be L.

We have been given that (x1, y1) = (7,4).

We have also been given that the line L is perpendicular to another line L1 whose equation is given by: y = 3x - 1

Hence, the slope of line L1 is m1 = 3.

L ⊥ L1 ⇒ m × m1 = -1

⇒ 3m = -1

⇒ m = -1/3

Hence, the equation of required line is given by:

⇒ y - y1 = m (x - x1)

⇒ y - 4 = -1/3 (x - 7)

⇒ 3(y - 4) = -(x - 7)

⇒ 3y - 14 = 7 - x

⇒ 3y = 14 + 7 - x

⇒ 3y = - x + 21

⇒ y = -1/3 x + 7.

This is the required equation of line.

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Answer 2

EXPLANATION.

Equation of the line that perpendicular to the line y = 3x - 1.

Passes through the point (7,4).

As we know that,

Slope of perpendicular line : ax + by + c is b/a.

Slope of line : 3x - y - 1 = 0 is -1/3.

Slope : m = -1/3.

Formula of equation of line.

⇒ (y - y₁) = m(x - x₁).

Using this formula in this question, we get.

⇒ (y - 4) = (-1/3)(x - 7).

⇒ 3(y - 4) = -(x - 7).

⇒ 3y - 12 = - x + 7.

⇒ 3y - 12 + x - 7 = 0.

⇒ x + 3y - 19 = 0.

⇒ x + 3y = 19.

∴ Equation of line is : x + 3y = 19.

                                                                                                                 

MORE INFORMATION.

Different forms of the equation of straight line.

(1) Slope - Intercept form : y = mx + c.

(2) Slope point form : The equation of a line with slope m and passing through a point (x₁, y₁) is : (y - y₁) = m(x - x₁).

(3) Two point form : (y - y₁) = [(y₂ - y₁)/(x₂ - x₁)](x - x₁).

(4) Intercept form : x/a + y/b = 1.

(5) Normal (perpendicular) form of line : x cosα + y sinα = p.

(6) Parametric form (distance form) : (x - x₁)/cosθ = (y - y₁)/sinθ = r.