If the tangent at the point $\left(x_1,y_1\right)$ on the curve $y=x^3+3x^2+5$ passes through the origin then $\left(x_1,y_1\right)$ does not lie on the curve

Tangent at  $\left(x_1{y}_1\right)$ to the curve $y=x^3+3x^2+5$ is $y-y_1=\left(3x_1^2+6x_1\right)\left(x-x_1\right)$ passing through origin

$-y_1=\left(3x_1^2+b{x}_1\right)\left(-x_1\right)\Rightarrow y_1=3x_1^3+6x_1^2---\left(1\right)$

$\left(x_1, y_1\right)$ lie on the curve 

$y=x^3+3x^2+5\Rightarrow y_1=x_1^3+3x_1^2+5---\left(2\right)$

From (1) & (2) 

$\Rightarrow 2y_1=3x_1^2+15\Rightarrow y_1=\frac{3}{2}{x}_1^2+\frac{15}{2}\Rightarrow$$y=\frac{3}{2}{x}^2+\frac{15}{2}$

Does not intersect $\frac{x}{3}-y^2=2$