Question

If $y=mx+5$ is a tangent to $x^3{y}^3=a{x}^3+b{y}^3$ at point $\left(1,2\right),$ then the value of $a$ is equal to

• A. $\frac{9}{5}$
• B. $\frac{16}{5}$
• C. $\frac{9}{4}$
• D. $\frac{18}{7}$

Answer 1

Answer: (B)

$P\left(1,2\right)$ lies on the curve and line, hence
$L\to 2=m+5\Rightarrow m=-3$
$Curve\to 8=a+8b$
Also, $\frac{dy}{dx}$ at $P\left(1,2\right)$ for the curve should be $m=-3,$
i.e. $3x^2{y}^3+x^3.3y^2{y}^{{}^{\prime }}=3a{x}^2+3b{y}^2{y}^{{}^{\prime }}$
$\Rightarrow 24+12\left(-3\right)=3a+12b\left(-3\right)$
$\Rightarrow -12=3a-36b$
$\Rightarrow a-12b=-4$ & $a+8b=8$
$-20b=-12$
$\Rightarrow b=\frac{3}{5}$ and $a=\frac{16}{5}$