Given the line 3x + 4y + 10 = 0 find the angle wich it makes with the axis of x  an its intercepts upon the axis.

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

3x+4y+10 = 0

To find the angle the line makes with x axis.

We rewrite the given equation in the slope intrecept form like y = mx+c, where m is the slope of the line with x axis.

For this , we subtract 3x+10 from both sides of the given equation.

4y = 0-(3x+10)

Divide by 4:

y =  -(1/4)(3x+10)

y = (-3/4)x-10/4. Now if this equation and y = mx+c are identical, then m= -3/4 and c= -10/4 = -5/2.

So the slope of the given equation = -3/4.

But slope is the tangent of the angle that the line makes with x axis.

Therefore , tanx = (-3/4).

The intercept of x axis is  got by putting y = 0  and solving for x in 3x+4y +10 = 0. So 3x +4*0+10 = 0. So 3x = -10. So x = -10/3.

Therefore tanx = -(3/4). x = arctan (-3/4) =  -36.86989 degree nearly.

 

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll write the equation of the line into the standard form.

y = m x + n, where m is the slope and n is the y intercept.

We know that m = tan a, where a is the angle made by the line with the axis of X.

We'll re-write the equation of the line:

y = tan a*x + n

We'll put the equation 3x + 4y + 10 = 0 in the standard form. For this reason, we'll isolate 4y to the left side. We'll subtract 3x + 10 both sides:

4y = -3x - 10

We'll divide by 4:

y = -3x/4 - 10/4

y = -3x/4 - 5/2

The angle made by the line with the axis of X is m = tan a.

We'll identify m = -3/4

tan a = -3/4

a = arctan (-3/4) + k*pi

a = - arctan (3/4) + k*pi

The x intercept of the line is found when y = 0

But y = -3x/4 - 5/2

-3x/4 - 5/2 = 0

We'll add 5/2 both sides:

-3x/4 = 5/2

We'll cross multiply and we'll get:

-6x = 20

We'll divide by -6:

x = -20/6

x = -10/3

So the line is intercepting x axis in the point (-10/3 , 0).

When the line is intercepting y axis, x = 0

So, y intercept is n = -5/2.

So the line is intercepting y axis in the point (0 , -5/2).

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