Problem Analysis:

The problem involves a curve $y^2 + 2x = 13$ and a line $2y + x = k$, where $k$ is a constant. There are two parts:

1. Part (i): When $k = 8$, find the coordinates of the points of intersection between the line and the curve.

2. Part (ii): Determine the value of $k$ for which the line is tangent to the curve.

Solution to Part (i):

Step 1: Substitute the equation of the line into the curve

From the line $2y + x = 8$, rearrange to express $x$ in terms of $y$: $x = 8 - 2y.$ Substitute this into the curve equation $y^2 + 2x = 13$: $y^2 + 2(8 - 2y) = 13.$ Simplify: $y^2 + 16 - 4y = 13,$ $y^2 - 4y + 3 = 0.$

Step 2: Solve the quadratic equation for $y$

Factorize: $y^2 - 4y + 3 = (y - 3)(y - 1) = 0.$ Thus: $y = 3 \quad \text{or} \quad y = 1.$

Step 3: Find corresponding $x$-coordinates

Using $x = 8 - 2y$:

• When $y = 3$: $x = 8 - 2(3) = 8 - 6 = 2.$
• When $y = 1$: $x = 8 - 2(1) = 8 - 2 = 6.$

Final Answer for Part (i):

The points of intersection are: $(2, 3) \quad \text{and} \quad (6, 1).$

Solution to Part (ii):

For the line to be tangent to the curve, the quadratic equation resulting from substitution must have exactly one solution (i.e., the discriminant must be zero).

Step 1: General substitution

From the line $2y + x = k$, rearrange to express $x$: $x = k - 2y.$ Substitute this into $y^2 + 2x = 13$: $y^2 + 2(k - 2y) = 13.$ Simplify: $y^2 + 2k - 4y = 13,$ $y^2 - 4y + (2k - 13) = 0.$

Step 2: Tangency condition

For tangency, the discriminant ($\Delta$) of the quadratic equation $y^2 - 4y + (2k - 13) = 0$ must be zero: $\Delta = b^2 - 4ac,$ where $a = 1$, $b = -4$, and $c = 2k - 13$. Compute: $\Delta = (-4)^2 - 4(1)(2k - 13).$ $\Delta = 16 - 8k + 52.$ $\Delta = 68 - 8k.$ Set $\Delta = 0$: $68 - 8k = 0.$ Solve for $k$: $8k = 68,$ $k = 8.5.$

Final Answer for Part (ii):

The value of $k$ for which the line is tangent to the curve is: $k = 8.5.$

Summary:

1. (i): Points of intersection for $k = 8$ are $(2, 3)$ and $(6, 1)$.
2. (ii): The value of $k$ for tangency is $k = 8.5$.

Would you like a step-by-step breakdown or clarification on any part of this solution?

Five Related Questions:

1. What is the geometric significance of a tangent line to a curve?
2. How do you find the point of tangency once $k$ is known?
3. Can you solve similar problems for cubic equations?
4. What happens if $\Delta > 0$ or $\Delta < 0$ for intersection problems?
5. How would this problem change if the curve equation were a circle?

Tip: Always check the discriminant ($\Delta$) to confirm tangency conditions in quadratic equations!