Part (a): Find the coordinates of the points of intersection when $k = 2$

The equation of the curve is:

$xy = 12 + x$

The equation of the line is:

$y = kx - 9$

When $k = 2$, the line equation becomes:

$y = 2x - 9$

Substitute $y = 2x - 9$ into $xy = 12 + x$:

$x(2x - 9) = 12 + x$

Expand and simplify:

$2x^2 - 9x = 12 + x$

$2x^2 - 10x - 12 = 0$

Simplify further by dividing through by 2:

$x^2 - 5x - 6 = 0$

Factorize:

$(x - 6)(x + 1) = 0$

Thus, $x = 6$ or $x = -1$.

Now find the corresponding $y$-coordinates:

• For $x = 6$, substitute into $y = 2x - 9$: $y = 2(6) - 9 = 12 - 9 = 3$ So, one intersection point is $(6, 3)$.

• For $x = -1$, substitute into $y = 2x - 9$: $y = 2(-1) - 9 = -2 - 9 = -11$ So, the other intersection point is $(-1, -11)$.

Final Answer for Part (a):

The points of intersection are: $(6, 3) \quad \text{and} \quad (-1, -11).$

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Part (b): Find the set of values of $k$ for which the line does not intersect the curve

Substitute $y = kx - 9$ into the curve equation $xy = 12 + x$:

$x(kx - 9) = 12 + x$

Expand and simplify:

$kx^2 - 9x = 12 + x$

$kx^2 - 10x - 12 = 0$

This is a quadratic equation in $x$, and the discriminant ($\Delta$) determines whether the line intersects the curve. The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is:

$\Delta = b^2 - 4ac$

For $kx^2 - 10x - 12 = 0$:

• $a = k$
• $b = -10$
• $c = -12$

The discriminant is:

$\Delta = (-10)^2 - 4(k)(-12)$

$\Delta = 100 + 48k$

For the line to not intersect the curve, the discriminant must be negative:

$100 + 48k < 0$

Solve for $k$:

$48k < -100$

$k < -\frac{100}{48}$

$k < -\frac{25}{12}$

Final Answer for Part (b):

The line does not intersect the curve when:

$k < -\frac{25}{12}.$

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Questions for Further Exploration:

1. How would the solution change if the line equation included a different constant term instead of $-9$?
2. What happens to the points of intersection if $k = 0$?
3. How can we determine the range of $y$-values for the curve $xy = 12 + x$?
4. How would you graphically confirm the set of $k$ values where no intersection occurs?
5. How does the discriminant's value relate to the type of intersection (tangency, two points, or no intersection)?

Tip:

Always analyze the discriminant carefully when studying the intersection of quadratic equations with other curves or lines—it provides a powerful method to determine interaction.