SOLUTION: what is the point {{{P}}} such that {{{abs(AP)/abs(AB)=r}}} for A = (4, -2), B = (-2, -5), {{{r = 2/3}}} where {{{abs(AP)}}} denotes the length of the line segment {{{AP}}}
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-> SOLUTION: what is the point {{{P}}} such that {{{abs(AP)/abs(AB)=r}}} for A = (4, -2), B = (-2, -5), {{{r = 2/3}}} where {{{abs(AP)}}} denotes the length of the line segment {{{AP}}}
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You can put this solution on YOUR website! The solution is not unique. I am assuming that you are only concerned with real coordinates on the plane. Thus we first calculate the length of
Now we know that P = ( x, y ) and thus .
Next we inspect the expression: . We plug in and and multiply by to yield . Now we square both sides and plug in the expression for . Thus we get the equation . Then we subtract 20 to yield .
Now is a tricky part. We have two unknowns and and only one equation. Thus there are infinitely many solutions. Thus we can choose to be anything, then solve for . Set , , and , then our equation simply looks like the standard quadratic Then we use the quadratic equation to yield . Here I remark that only positive things can be inside the square root. Thus we must choose such that implies implies . Now we factor the quadratic using the quadratic formula: . We plug this into the inequality and note that is a solution to yield that This implies that has solutions in the set [, ].
One particular solution occurs when x = 0. Thus in this case, and we can take the positive version of the square root to yield
Check the answer:
.
Thus
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