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in Co-ordinate geometry by (50 points)
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If 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 :

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Point (x1, y1) lies on the curve

\(y = x^3 + 3x^2 + 5\)

\(\therefore y_1 = {x_1}^3 + 3{x_1}^2 + 5\)    ....(1)

\(\frac{dy}{dx} = 3x^2 + 6x\)

\(\therefore \left(\frac{dy}{dx}\right) = 3x^2 + 6x\)

Tangent is passes through origin.

∴ Equation of tangent is

\(y - 0 = \left(\frac{dy}{dx}\right) _{(x_1, y_1)} (x - 0)\)

⇒ \(y = (3x^2 + 6x )(x - 0)\)

⇒ \(y = 3x^3 + 6x^2\)       .....(2)

Point (x1, y1) lies on curve (2).

\(\therefore y_1 = 3{x_1}^3 + 6{x_1}^2\)

\(= 3({x_1}^3 + 2{x_1}^2) \)

\(=3({y_1} - {x_1}^2 - 5) \)

\(= 3y_1 - 3{x_1}^2 - 15\)

⇒ \(3{x_1}^2 = 2y_1 - 15\)

Hence, (x1, y1) lies on 3x2 = 2y - 15.

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