Question find the point on the line 2x+3y+4=0 which is closest to the point(4,4).

&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/2Given equation of line is 2x+3y+4=0&#160; and the given &#160;point is (4,4) Note that geometrically the pointwhich is closest to (4,4) and on the given line is must be foot of perpendicular from (4,4).Let (a,b ) be coordinate of foot of perpendicular .The the product of slope of given line and line perpendicular to it must be -1.Since slope of given line can be extracted by ,$$\mathrm{{2}{x}+{3}{y}+{4}={0}}$$$$\mathrm{{y}=−\frac{{2}}{{3}}{x}+\frac{{4}}{{3}}}$$&#160;&#160; On compairing with standard equation of line that is y=mx+c We have slope m=-2/3Now slope of line perpendicular to it is \begin{align*} \mathrm{-\frac{{2}}{{3}}×{m}} &= \mathrm{-{1}}\\[3pt]\mathrm{{m}} &= \mathrm{\frac{{3}}{{2}}} \end{align*}Now we have to find equation of line passing through (4,4) and having slope 3/2.\( \begin{align*} \mathrm{\frac{{{y}-{4}}}{{{x}-{4}}}} &= \mathrm{\frac{{3}}{{2}}}\\[3pt]\mathrm{{2}{y}-{8}} &= \mathrm{{3}{x}-{12}}\\[3pt]\mathrm{{3}{x}-{2}{y}} &= \m ... See the full answer