Question 1004906
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 {{{abs(AB) = sqrt( (A_1-B_1)^2 + (A_2-B_2)^2 ) = sqrt ( (4 - (-2))^2 + ( (-2) - (-5) )^2 ) = sqrt( 36 + 9 ) = sqrt( 45 ) = sqrt( 3^2 * 5 ) = sqrt(3^2) * sqrt(5) = 3*sqrt(5) }}}

Now we know that P = ( x, y ) and thus {{{ abs(AP)^2 = (A_1-x)^2 + (A_2-y)^2 = (4-x)^2 + (-2-y)^2 = ( 16 - 8 x + x^2 ) + ( 4 + 4y + y^2 )}}}.

Next we inspect the expression: {{{ abs(AP)/abs(AB) = r }}}. We plug in {{{r}}} and {{{abs(AB)}}} and multiply by {{{abs(AB)}}} to yield {{{ abs(AP) = (2/3)*(3*sqrt(5) ) = 2 * sqrt(5) }}}. Now we square both sides and plug in the expression for {{{abs(AP)^2}}}. Thus we get the equation {{{ (20 - 8 x + x^2) + (4) y + y^2 = 20 }}}. Then we subtract 20 to yield {{{ (8 x + x^2) + (4) y + y^2 = 0 }}}.

Now is a tricky part. We have two unknowns {{{x}}} and {{{y}}} and only one equation. Thus there are infinitely many solutions. Thus we can choose {{{x}}} to be anything, then solve for {{{y}}}. Set {{{ a = 1}}}, {{{ b= 4}}}, and {{{c=-8x + x^2}}}, then our equation simply looks like the standard quadratic {{{a*y^2 + b*y + c = 0}}} Then we use the quadratic equation to yield {{{ y = ( -b +- sqrt( b^2 - 4ac ) ) / 2a = ( -4 +- sqrt( 16 - 4c ) ) / 2 =  (-4 +- 2sqrt( 4 - c ) ) / 2 = -2 +- sqrt( 4 - c )}}}. Here I remark that only positive things can be inside the square root. Thus we must choose {{{x}}} such that {{{ c <= 4}}} implies {{{ - 8x + x^2 <= 4}}} implies {{{ -4 - 8x + x^2 <=0 }}}. Now we factor the quadratic using the quadratic formula: {{{-4 - 8x + x^2 = ( x - (8+sqrt(64+16))/2 )(x-(8-sqrt(64+24))/2 ) = ( x - (8+2sqrt(20))/2 )(x-(8-2sqrt(20))/2 ) =  ( x - (4+sqrt(20)) )(x-(4-sqrt(20)) ) }}}. We plug this into the inequality and note that {{{x=0}}} is a solution to yield that {{{ ( x - (4+sqrt(20)) )(x-(4-sqrt(20)) ) <= 0 }}} This implies that {{{x}}} has solutions in the set [{{{4-sqrt(20)}}}, {{{4+sqrt(20)}}} ]. 

One particular solution occurs when x = 0. Thus in this case, {{{c=0}}} and we can take the positive version of the square root to yield {{{ y = -2 + sqrt(4) }}}

Check the answer:

{{{ abs(AP)^2 = 16 + (- sqrt(4))^2 = 16  + 4 = 20 }}}.

Thus {{{ abs(AP) / abs(AB) = sqrt(20) / (3*sqrt(5)) = (2*sqrt(5))/(3*sqrt(5)) = 2/3 }}}