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Q.
the tangent at the point $\left(x_{1}, y_{1}\right)$ on the curve $y=x^{3}+3 x^{2}+5$ passes through the origin, then $\left( x _{1}, y _{1}\right)$ does NOT lie on the curve :
The tangent at $\left(x_{1}, y_{1}\right)$ to the curve
$y=x^{3}+3 x^{2}+5$
$y-y_{1}=\left(3 x_{1}^{2}+6 x_{1}\right)\left(x-x_{1}\right) $
passing through origin
$-y_{1}=\left(3 x_{1}^{3}+6 x_{1}\right)\left(-x_{1}\right) $
$y_{1}=\left(3 x_{1}^{3}+6 x_{1}^{2}\right)\,\,\,$-----(1)
And $\left( x _{1}, y _{1}\right)$ lies on the curve
$y=x^{3}+3 x^{2}+5$
$y_{1}=x_{1}^{3}+3 x_{1}^{2}+5 $----(2)
From equation (1) and (2)
$2 y_{1}=3 x_{1}^{2}+\frac{15}{2}$
Hence the equation of curve $y=\frac{3}{2} x^{2}+\frac{15}{2}$
This curve does not intersect $\frac{x}{3}-y^{2}=2$