Question

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 :

• A. $x^2+\frac{y^2}{81}=2$
• B. $\frac{y^2}{9}-x^2=8$
• C. $y=4x^2+5$
• D. $\frac{x}{3}-y^2=2$

Answer 1

Answer: (D)

The tangent at $\left(x_1,y_1\right)$ to the curve
$y=x^3+3x^2+5$
$y-y_1=\left(3x_1^2+6x_1\right)\left(x-x_1\right)$
passing through origin
$-y_1=\left(3x_1^3+6x_1\right)\left(-x_1\right)$
$y_1=\left(3x_1^3+6x_1^2\right)$-----(1)
And $\left(x_1,y_1\right)$ lies on the curve
$y=x^3+3x^2+5$
$y_1=x_1^3+3x_1^2+5$----(2)
From equation (1) and (2)
$2y_1=3x_1^2+\frac{15}{2}$
Hence the equation of curve $y=\frac{3}{2}{x}^2+\frac{15}{2}$
This curve does not intersect $\frac{x}{3}-y^2=2$