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