Respuesta :
The statement about the line that passes through (-2, 0) and (2, -4) that are true are:
Option B:The line intersects the y-axis at (0, −2).
Option C: The equation of the line is y = −x − 2.
Option D: The line intersects the x-axis at (−2, 0).
Solution:
Given, two points are (-2, 0) and (2, -4)
We have to select the options that states true about line that passes through given two points.
Now, let us find the line equation that passes through given two points using point slope form.
[tex]\mathrm{y}-\mathrm{y}_{1}=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right) \text { where } \mathrm{m} \text { is slope and }\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) \text { is a point on the line. }[/tex]
Now slope "m" is given as:
[tex]\mathrm{m}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{0-(-4)}{-2-2}=\frac{4}{-4}=-1[/tex]
Then, line equation = y – 0 = -1(x – (-2))
y = -1(x + 2)
y = -x – 2 --- equation 1
Now let us check options.
Option a) slope of line is 1
We know that slope of our line is -1 so this option is wrong.
Option b) line intersects y – axis at (0, -2)
When line meets y – axis x becomes 0 ⇒ 0 + y + 2 = 0 ⇒ y = -2 so point is (0, -2).
This option is right.
Option c) equation of line is y = -x – 2
The given equation is x + y + 2 = 0 ⇒ y = -x – 2 . so this option is correct.
Option d) line intersects at (-2, 0)
When line meets x – axis y becomes 0 ⇒ x + 0 + 2 = 0 ⇒ x = -2. So point is (-2, 0).
This option is right.
Hence, options b, c, d are correct.
The correct options are B, C, and D.
- The line intersects the y-axis at (0, −2).
- The equation of the line is y = −x − 2.
- The line intersects the x-axis at (−2, 0).
Given the coordinate points (−2, 0) and (2, −4)
Get the slope:
[tex]m =\frac{y_2-y_1}{x_2-x_1} \\m=\frac{-4-0}{2-(-2)}\\m=\frac{-4}{4}\\m = -1[/tex]
Hence the slope is -1 falsifying option A
Get the y-intercept
Using the point (-2, 0) and m = -1
[tex]0 = -1(-2) + b\\0 = 2 + b\\b = -2\\[/tex]
Hence the y-intercept is -2. This shows that the line intersects the y-axis at (0, −2). Option B is true
Get the equation of the line.
Recall that y = mx+ b
y = -x + (-2)
y = -x - 2
Hence the equation of the line is y = −x − 2. Option C is correct.
Get the x-intercept.
if y = 0
0 = -x - 2
x = -2
Hence the line intersects the x-axis at (−2, 0). option D is correct.
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