Question

Find the equations to the straight lines passing through $(x^{\prime },y^{\prime })$ and respectively perpendicular to the straight lines
$x{x}^{\prime }+y{y}^{\prime }=a^2$,
$\frac{x{x}^{\prime }}{a^2}+\frac{y{y}^{\prime }}{b^2}=1$,
and $x^{\prime }y+x{y}^{\prime }=a^2$

Answer 1

(1) $x{x}^{\prime }+y{y}^{\prime }=a^{2}y{y}^{\prime }=-x{x}^{\prime }+a^{2}y=-\frac{x^{\prime }}{y^{\prime }}x+\frac{a^2}{y^{\prime }}$

Slope $=m=-\frac{x^{\prime }}{y^{\prime }}$

Let slope of other line be $m^{\prime }$

$m{m}^{\prime }=-1-\frac{x^{\prime }}{y^{\prime }}{m}^{\prime }=-1m^{\prime }=\frac{y^{\prime }}{x^{\prime }}$

Equation of perpendicular line

$y-y^{\prime }=\frac{y^{\prime }}{x^{\prime }}(x-x^{\prime })y{x}^{\prime }-y^{\prime }{x}^{\prime }=y^{\prime }x-y^{\prime }{x}^{\prime }y{x}^{\prime }-x{y}^{\prime }=0$

$-------------------------------------------$

(2) $\frac{x{x}^{\prime }}{a^2}+\frac{y{y}^{\prime }}{b^2}=1$

$\frac{y{y}^{\prime }}{b^2}=-\frac{x{x}^{\prime }}{a^2}+1y=-\frac{b^2{x}^{\prime }}{a^2{y}^{\prime }}x+\frac{b^2}{y^{\prime }}$

Slope of line $=m=-\frac{b^2{x}^{\prime }}{a^2{y}^{\prime }}$

Let the slope of perpendicular line be $m^{\prime }$

$m{m}^{\prime }=-1-\frac{b^2{x}^{\prime }}{a^2{y}^{\prime }}{m}^{\prime }=-1\Rightarrow m^{\prime }=\frac{a^2{y}^{\prime }}{b^2{x}^{\prime }}$

Equation of perpendicular line is

$y-y^{\prime }=\frac{a^2{y}^{\prime }}{b^2{x}^{\prime }}(x-x^{\prime })b^2{x}^{\prime }y-b^2{x}^{\prime }{y}^{\prime }=a^2{y}^{\prime }x-a^2{y}^{\prime }{x}^{\prime }{a}^2{y}^{\prime }x-b^2{x}^{\prime }y=a^2{x}^{\prime }{y}^{\prime }-b^2{y}^{\prime }{x}^{\prime }{a}^2{y}^{\prime }x-b^2{x}^{\prime }y=(a^2-b^2)x^{\prime }{y}^{\prime }$

$----------------------------------------------$

(3) $x^{\prime }y+x{y}^{\prime }=a^2$
$x^{\prime }y=-y^{\prime }x+a^{2}y=-\frac{y^{\prime }}{x^{\prime }}x+\frac{a^2}{x^{\prime }}$

Slope of line $=m=-\frac{y^{\prime }}{x^{\prime }}$

Let slope of perpendicular line be $m^{\prime }$

$m{m}^{\prime }=-1-\frac{y^{\prime }}{x^{\prime }}{m}^{\prime }=-1\Rightarrow m^{\prime }=\frac{x^{\prime }}{y^{\prime }}$

Equation of perpendicular line is

$y-y^{\prime }=\frac{x^{\prime }}{y^{\prime }}(x-x^{\prime })y^{\prime }y-{y^{\prime }}^2=x^{\prime }x-{x^{\prime }}^2{y}^{\prime }y-x^{\prime }x={y^{\prime }}^2-{x^{\prime }}^2$