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# Analytic Geometry

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Feb 19, 2024

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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=21​x+4y=−2x+15​\$

Substituting the first equation into the second equation, we get: \$21​x+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).

Feb 21, 2024