+193 Here are the steps for problem 2:
Find the equation of the perpendicular line passing through A: The slope of the given line is 21, and its negative reciprocal is −2.
Thus, the equation of the perpendicular line passing through A will be of the form y=−2x+b. We can plug A's coordinates into this equation to solve for b: $1=−2(7)+b⟹b=15.$
Therefore, the equation of the perpendicular line is y=−2x+15.
Find the intersection point of the two lines: The intersection point of the two lines will be the midpoint of the segment connecting A and its reflection B.
To find it, we can solve the system of equations: ${y=21x+4y=−2x+15$
Substituting the first equation into the second equation, we get: $21x+4=−2x+15.$
Solving for x, we find x=35. Substituting this value back into either equation, we find y=317.
Therefore, the intersection point is (35,317).
Find the reflection point B: Since the reflection point B is the same distance from A as the intersection point is from A, we can find B by calculating the vector from A to the intersection point and multiplying it by 2.
The vector from A to the intersection point is (35−7,317−1)=(−316,314).
Multiplying this vector by 2, we get (−332,328). Finally, adding this vector to A's coordinates gives us the coordinates of B: (7,1) + (-32/3, 28/3) = (1/3, 31/3).
Therefore, B = (1/3, 31/3).
