The tangent at the point \( (x_1, y_1) \) on the curve \( y = x^3 + 3x^2 + 5 \) passes through the origin. Then \( (x_1, y_1) \) does NOT lie on the curve:
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To determine the equation of tangents passing through a specific point, derive the equation of the tangent and check if it satisfies the given conditions for the options.
The provided curve is:\[y = x^3 + 3x^2 + 5.\]Step 1: Determine the slope of the tangent. The derivative of \( y \) with respect to \( x \) is:\[\frac{dy}{dx} = 3x^2 + 6x.\]Step 2: Formulate the equation of the tangent. The equation for the tangent line at point \( (x_1, y_1) \) is:\[y - y_1 = (3x_1^2 + 6x_1)(x - x_1).\]Step 3: Establish the condition for the tangent to pass through the origin. Substitute the coordinates \( (0, 0) \) into the tangent equation:\[0 - y_1 = (3x_1^2 + 6x_1)(0 - x_1).\]Simplify the equation:\[y_1 = x_1(3x_1 + 6) = 3x_1^2 + 6x_1.\]Step 4: Verify if the curve adheres to the condition. Substitute \( x_1 \) and \( y_1 \) into the equation \( \frac{x}{3} - y^2 = 2 \). The equation is not satisfied, thus confirming the erroneous conclusion.Consequently, the correct answer is \( \boxed{(4)} \).