Question 785377
Question #1.


The line contains some general point, (x, 4x+5).  The distance from this general point to one of the given points must be equal to the distance from the general point to the other given point.  Setup the distance formula expressions.


From the line to (-9,7):  {{{sqrt((x-(-9))^2+(4x+5-7)^2)}}}
{{{sqrt((x+9)^2+(4x-2)^2)}}}
{{{sqrt(x^2+18x+81+16x^2-16x+4)}}}
{{{sqrt(17x^2+2x+85)}}}



From the line to (-8,3):   {{{sqrt((x-(-8))^2+(4x+5-3)^2)}}}
{{{sqrt((x+8)^2+(4x+2)^2)}}}
{{{sqrt(x^2+18x+64+16x^2+16x+4)}}}
{{{sqrt(17x^2+34x+68)}}}


These two expressions, equated and then both sides squared, give
{{{17x^2+2x+85=17x^2+34x+68}}}
{{{2x+85=34x+68}}}
{{{85=32x+68}}}
{{{86-68=32x}}}
{{{32x=18}}}
{{{x=18/32=highlight(9/16=x)}}}


Use the line equation again for finding y.
{{{y=(9/16)*4+5}}}
{{{y=9/4+5}}}
{{{y=(9+20)/4=29/4}}}
{{{highlight(y=7&1/4=29/4)}}}
'
Point solution is (9/16, 29/4)