Mathematics - Application of Derivative Question with Solution | TestHub

Slope of tangent at $P\left(x_1,y_1\right)$ is $-\frac{x_1^2}{y_1^2}$, so points on the tangent are $\left(x_1+t{y}_1^2,y_1-t{x}_1^2\right)$. Let $h=x_1+t{y}_1^2,k=y_1-t{x}_1^2$ where $Q(h,k)$ lies on $x^3+y^3=a^3$: ${\left(x_1+t{y}_1^2\right)}^3+{\left(y_1-t{x}_1^2\right)}^3=a^3$. Using $x_1^3+y_1^3=a^3$ and $t\ne 0$: $3x_1{y}_1+t\left(y_1^3-x_1^3\right)=0\Rightarrow t=\frac{3x_1{y}_1}{x_1^3-y_1^3}$ so $A=\frac{h}{x_1}=1+\frac{3y_1^3}{x_1^3-y_1^3}$, $B=\frac{k}{y_1}=1-\frac{3x_1^3}{x_1^3-y_1^3}$, $B-A=\frac{3a^3}{y_1^3-x_1^3}$ and $1-B=\frac{3x_1^3}{x_1^3-y_1^3}$ $\Rightarrow \frac{1-B}{B-A}=-\frac{x_1^3}{a^3}\Rightarrow x_1^3=\frac{a^3(1-B)}{A-B}$