Straight Lines

To find the length of the x-intercept made by the pair of lines given by the equation , we first determine the points where the lines intersect the x-axis. This happens when . By substituting into the equation, we simplify:

Dividing the entire equation by 2 gives:

This is a quadratic equation in the form , where , , and . To find the roots, use the quadratic formula:

Substitute the values of , , and :

This results in:

and

The x-intercepts are and . The distance between these intercepts is calculated as the absolute difference:

Distance =

Thus, the length of the x-intercept made by the pair of lines is .

The given equation of the pair of lines is .
We have the general form of the equation of pair of lines as .
Comparing, we identify , (hence ), and .

The condition for the slope of one of the lines is given by the equation:
.
Substituting :
.

Simplifying the equation:
.
To find the value of , solve the equation:
.
Squaring both sides:
.
This equation can get quite complex, so consider backtracking if calculations hit an impasse. Generally for simplicity, you'll find where the existence condition returns smoothly.

Now, for where is the angle between the lines, use:
.
Substitute known values to find
.
This angle turns to when lines are perpendicular.
Perpendicular lines imply which means .

Thus, the correct answer is:

Step 1: Finding the equation of the line The equation of the line through and : Simplifying, Step 2: Condition for tangency The given curve is . For tangency, we equate slopes: Solving for , we get:

Let .The midpoint of is , and this point lies on the line , so This simplifies to . [asy] unitsize(2 cm); pair A, B, C, D, E; A = (1,
-2); B = (2,15); C = (8,
-4); D = (A + B)/2; E = (A + C)/2; draw(A
-
-B
-
-C
-
-cycle); draw(D
-
-D + 3*(D
- E)); draw(E
-
-E + 3*(E
- D)); label(" ", A, SE); label(" ", B, N); label(" ", C, SE); label(" ", D, NE); label(" ", E, S); [/asy] Also, the slope of is , and the slope of the line is 1, so This simplifies to . Solving the system and , we find . Let .The midpoint of is , and this point lies on the line , so This simplifies to . Also, the slope of is , and the slope of the line is , so This simplifies to . Solving the system and , we find . Then the slope of is so the equation of line is of the form which simplifies to .