1. **State the problem:** We are given the line equation $y = 4x - 1$ and the conic equation $x^2 + 2x + y^2 = k$. We want to analyze the relationship between these two equations. 2. **Substitute $y$ from the line into the conic:** Replace $y$ in the conic equation with $4x - 1$: $$x^2 + 2x + (4x - 1)^2 = k$$ 3. **Expand the squared term:** $$(4x - 1)^2 = 16x^2 - 8x + 1$$ 4. **Rewrite the conic equation:** $$x^2 + 2x + 16x^2 - 8x + 1 = k$$ 5. **Combine like terms:** $$17x^2 - 6x + 1 = k$$ 6. **Rearrange to standard quadratic form:** $$17x^2 - 6x + (1 - k) = 0$$ 7. **Interpretation:** This quadratic in $x$ represents the intersection points of the line and the conic for a given $k$. The number of real solutions depends on the discriminant: $$\Delta = (-6)^2 - 4 \cdot 17 \cdot (1 - k) = 36 - 68 + 68k = 68k - 32$$ 8. **Condition for intersection:** - If $\Delta > 0$, two distinct intersection points. - If $\Delta = 0$, one tangent point. - If $\Delta < 0$, no real intersection. 9. **Solve for $k$ to find intersection conditions:** $$68k - 32 \geq 0 \implies k \geq \frac{32}{68} = \frac{8}{17}$$ **Final answer:** The line $y=4x-1$ intersects the conic $x^2 + 2x + y^2 = k$ if and only if $k \geq \frac{8}{17}$.