Question

Find the equation to the pair of straight lines joining the origin to the intersections of the straight line y = mx + c and the curve
$x^2+y^2=a^2.$
Prove that they are at right angles if
$2c^2=a^2(1+m^2).$

Answer 1

$x^2+y^2=a^{2}y=mx+c$

The equation of pair of straight line joining the origin is find by method of homogenisation. We make the coefficient of constant term of straight line $1$ and put it the equation of the curve so that degree of each term of the curve become $2$

$c=y-mx1=\frac{y-mx}{c}...\left(i\right)x^2+y^2=a^2{\left(1\right)}^2{x}^2+y^2=a^2{\left(\frac{y-mx}{c}\right)}^2{x}^2+y^2=\left(\frac{a^2}{c^2}\right)y^2+\left(\frac{a^2{m}^2}{c^2}\right)x^2-2\left(\frac{a^{2}m}{c^2}\right)xy(1-\frac{a^2{m}^2}{c^2})x^2+(1-\frac{a^2}{c^2})+2\left(\frac{a^{2}m}{c^2}\right)xy=0$

is the desired equation of the pair of straight lines.

Lines are prependicular if $a+b=0$

$\Rightarrow 1-\frac{a^2{m}^2}{c^2}+1-\frac{a^2}{c^2}=02=\frac{a^2{m}^2+a^2}{c^2}2c^2=a^2(1+m^2)$

Hence proved.