1. **Problem statement:** We have two quadratic equations: - Equation 1: $x^2 + mx + 15 = 0$ with roots $\alpha$ and $\beta$. - Equation 2: $x^2 + hx + k = 0$ with roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$. We need to find: (a) The value of $k$. (b) An expression for $h$ in terms of $m$. 2. **Recall the relationships for roots of a quadratic equation:** For an equation $x^2 + px + q = 0$ with roots $r_1$ and $r_2$: - Sum of roots: $r_1 + r_2 = -p$ - Product of roots: $r_1 r_2 = q$ 3. **Apply these to the first equation:** - Sum: $\alpha + \beta = -m$ - Product: $\alpha \beta = 15$ 4. **For the second equation, roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$:** - Sum of roots: $$\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta}$$ - Product of roots: $$\frac{\alpha}{\beta} \times \frac{\beta}{\alpha} = 1$$ 5. **Find $k$ (product of roots of second equation):** Since product of roots equals $k$, we have: $$k = 1$$ 6. **Find $h$ (negative sum of roots of second equation):** First, express $\alpha^2 + \beta^2$ in terms of $m$ and known values: $$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = (-m)^2 - 2 \times 15 = m^2 - 30$$ Sum of roots of second equation: $$\frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{m^2 - 30}{15}$$ Therefore, $$h = - \text{(sum of roots)} = - \frac{m^2 - 30}{15} = \frac{30 - m^2}{15}$$ **Final answers:** - (a) $k = 1$ - (b) $h = \frac{30 - m^2}{15}$