Let's break down and solve each part step by step as per the image: --- ### **Given:** - **Straight line:** \( y = mx + c \) - **Parabola:** \( y = x^2 \) - **Point \( R(3, 2) \):** Lies on the parabola. - **Points \( S \) and \( T \):** Points where the straight line intersects the parabola. - **Point \( V(-1, 4) \):** Given. - **Point \( Q(5, 0) \):** Given. Let's tackle each sub-question one by one: --- ### **2.1 Calculate \( m \) and \( c \), and hence write down the equation for \( \ell \).** #### **Explanation:** - The straight line \( \ell \) passes through \( R(3, 2) \) and \( Q(5, 0) \). - The equation of a straight line: \( y = mx + c \). - Use the two points to find \( m \) and \( c \). #### **Step-by-step:** 1. **Find the slope \( m \):** \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 2}{5 - 3} = \frac{-2}{2} = -1 \] So, \( m = -1 \). 2. **Substitute one point to find \( c \):** Using \( R(3, 2) \): \[ 2 = -1 \cdot 3 + c \implies 2 = -3 + c \implies c = 5 \] 3. **Equation of the line:** \[ y = -x + 5 \] --- ### **2.2 Find the equation of the parabola.** #### **Explanation:** From the sketch and information given, the parabola is \( y = x^2 \). --- ### **2.3 Find the maximum length of \( ST \). At what value of \( c \) will \( ST \) be at its longest?** #### **Explanation:** - \( S \) and \( T \) are the intersection points of the line \( y = -x + c \) (general form) with the parabola \( y = x^2 \). #### **Step-by-step:** 1. **Set \( x^2 = -x + c \):** \[ x^2 + x - c = 0 \] This quadratic in \( x \) has solutions: \[ x = \frac{-1 \pm \sqrt{1 + 4c}}{2} \] 2. **Distance \( ST \):** The \( x \)-coordinates are: \[ x_1 = \frac{-1 + \sqrt{1 + 4c}}{2},\quad x_2 = \frac{-1 - \sqrt{1 + 4c}}{2} \] Distance: \[ |x_1 - x_2| = \left| \frac{-1 + \sqrt{1 + 4c}}{2} - \frac{-1 - \sqrt{1 + 4c}}{2} \right| = \left| \frac{2\sqrt{1+4c}}{2} \right| = |\sqrt{1+4c}| \] For maximum length, \( c \) must be as large as possible while intersection points are real, i.e., \( 1 + 4c \geq 0 \implies c \geq -\frac{1}{4} \). As \( c \to \infty \), the length increases without bound, but the maximum length for a finite value occurs when the line just touches the parabola (i.e., discriminant zero), but that's when \( S \) and \( T \) coincide. So, for any real intersection, the length increases with \( c \). **But** for a practical answer, use the value of \( c = 5 \) from before: \[ \text{Length} = \sqrt{1+4 \cdot 5} = \sqrt{21} \] --- ### **2.4 What is the equation of the circle with centre \( (1, 1) \) and radius equal to the distance from \( P \) to the origin?** #### **Explanation:** Find distance from \( P \) to the origin and use in the standard circle equation. But \( P \) is not explicitly given. If we assume \( P \) is the vertex of the parabola, its coordinates are \( (0, 0) \). So, **distance from \( P \) to the origin** = \( 0 \). But perhaps \( P \) is another specific point, but since it's not specified, we can only write the equation in general: Let the distance be \( r \). Equation: \[ (x - 1)^2 + (y - 1)^2 = r^2 \] If \( P = (0,0) \), \( r = 0 \): \[ (x-1)^2 + (y-1)^2 = 0 \] --- ### **2.5 What is the equation of the hyperbola through the point \( V \)?** #### **Explanation:** A standard hyperbola centered at the origin: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \). It passes through \( V(-1, 4) \): \[ \frac{(-1)^2}{a^2} - \frac{4^2}{b^2} = 1 \] \[ \frac{1}{a^2} - \frac{16}{b^2} = 1 \] This is the equation in terms of \( a \) and \( b \). --- ### **2.6 Use the graphs to solve the inequality \( y < x^2 \) and \( y \leq -x + 5 \).** #### **Explanation:** We need the region **below** the parabola and **on or below** the line. - Find the intersection points as previously calculated, and shade the region **below the parabola** and **below the line** between those points. --- ## **Summary of Answers:** ### **2.1** - Slope \( m = -1 \), intercept \( c = 5 \) - **Equation of the line:** \(\boxed{y = -x + 5}\) ### **2.2** - **Equation of the parabola:** \(\boxed{y = x^2}\) ### **2.3** - **Maximum value of \( ST \):** \( \boxed{\sqrt{1+4c}} \) - For \( c = 5 \), \( \boxed{\sqrt{21}} \) - \( ST \) increases as \( c \) increases. ### **2.4** - **Equation of the circle:** \(\boxed{(x-1)^2 + (y-1)^2 = r^2}\), where \( r \) is the distance from \( P \) to the origin. ### **2.5** - **Equation of the hyperbola:** \(\boxed{\frac{1}{a^2} - \frac{16}{b^2} = 1}\) for \( V(-1, 4) \). ### **2.6** - The region that satisfies \( y < x^2 \) and \( y \leq -x + 5 \) is the area **below the parabola** and **below the line** between the intersection points. --- **If you need a specific part explained in more detail or a graph, let me know!**