Question

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:

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

Hint:

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.

Answer 1

The Correct Option is D

The given curve is: \[ y = x^3 + 3x^2 + 5. \] Step 1: Find the slope of the tangent. The derivative of \( y \) with respect to \( x \) is: \[ \frac{dy}{dx} = 3x^2 + 6x. \] Step 2: Equation of the tangent. The equation of the tangent line at the point \( (x_1, y_1) \) is: \[ y - y_1 = (3x_1^2 + 6x_1)(x - x_1). \] Step 3: 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: Check if the curve satisfies 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 incorrect answer. Therefore, the correct answer is \( \boxed{(4)} \).