Question

If the relation between length of sub-normal and length of sub-tangent at any point on the curve $b{y}^2=(x+a{)}^3$ is $p\left(SN\right)=q(ST{)}^2$, then $p:q=$

• A. $b:a$
• B. $a:b$
• C. $8b:27a$
• D. $8b:27$

Answer 1

The correct option is D $8b:27$

$b{y}^2=(a+x{)}^3$
Length of Sub-Tangent, $ST=ym$

Length of Sub-Normal, $SN=\frac{y}{m}$
$p\left(ym\right)=q{\left(\frac{y}{m}\right)}^2$
$\frac{p}{q}=\frac{y^2}{m^2\times ym}=\frac{y}{m^3}$ ...{i}

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

Using $y^{\prime }$ as $m$ in {i}, we get

$\frac{p}{q}=\frac{y\times (2by{)}^3}{\left(3\right(x+a{)}^2{)}^3}=\frac{8b^3{y}^4}{27(x+a{)}^6}$

$=\frac{8b^3}{27b^2}=\frac{8b}{27}$