Normal meets curve/axis — further geometry

The curve with equation \(y = x^2 + 2x\) passes through the origin, \(O\).

1. Find an equation for the normal to the curve at \(O\). [5]
2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again. [3]

A curve has the equation \(y = (\sqrt{x} - 3)^2\), \(x \geq 0\).

1. Show that \(\frac{dy}{dx} = 1 - \frac{3}{\sqrt{x}}\). [4]

The point \(P\) on the curve has \(x\)-coordinate 4.

1. Find an equation for the normal to the curve at \(P\) in the form \(y = mx + c\). [5]
2. Show that the normal to the curve at \(P\) does not intersect the curve again. [4]

A curve has the equation \(y = x + \frac{3}{x}\), \(x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate \(1\).

1. Show that the gradient of the curve at \(P\) is \(-2\). [3]
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = mx + c\). [3]
3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again. [4]

A curve has the equation \(y = x^3 - 5x^2 + 7x\).

1. Show that the curve only crosses the \(x\)-axis at one point. [4]

The point \(P\) on the curve has coordinates \((3, 3)\).

1. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [6]

The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).

1. Show that triangle \(OQR\), where \(O\) is the origin, has area \(28\frac{1}{8}\). [3]

Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_10}

1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C\((16, 0)\). [6]
2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]

\includegraphics{figure_2} Figure 2 shows the curve with equation \(y = 5 + x - x^2\) and the normal to the curve at the point \(P(1, 5)\).

1. Find an equation for the normal to the curve at \(P\) in the form \(y = mx + c\). [5]
2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again. [2]
3. Show that the area of the shaded region bounded by the curve and the straight line \(PQ\) is \(\frac{4}{3}\). [6]

Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_2}

1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C (16, 0). [6]
2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]

The point A has \(x\)-coordinate 5 and lies on the curve \(y = x^2 - 4x + 3\).

1. Sketch the curve. [2]
2. Use calculus to find the equation of the tangent to the curve at A. [4]
3. Show that the equation of the normal to the curve at A is \(x + 6y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again. [6]

The curve \(C\) has equation \(y = \text{f}(x)\), where $$\text{f}(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$ The point \(P\) is a stationary point on \(C\).

1. Calculate the \(x\)-coordinate of \(P\). [4]
2. Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. [2]

The point \(Q\) on \(C\) has \(x\)-coordinate 1.

1. Find an equation for the normal to \(C\) at \(Q\). [4]

The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).

1. Show that the \(x\)-coordinate of \(R\)
   1. satisfies the equation \(6 \ln x + x + \frac{2}{x} - 3 = 0\),
   2. lies between 0.13 and 0.14. [4]

A curve has the equation \(y = (2x + 3)e^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([-2, -1]\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3e^{x_n}}{e^{x_n} - 2}$$ with \(x_0 = -1\), to find \(x_1\), \(x_2\), \(x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [3]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

A curve has the equation \(y = (2x + 3)\mathrm{e}^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies between \(-2\) and \(-1\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3\mathrm{e}^{x_n}}{\mathrm{e}^{x_n} - 2},$$ with \(x_0 = -1\), to find \(x_1, x_2, x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [2]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

\includegraphics{figure_3} The curve \(C\) with equation \(y = 2e^x + 5\) meets the \(y\)-axis at the point \(M\), as shown in Fig. 3.

1. Find the equation of the normal to \(C\) at \(M\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]

This normal to \(C\) at \(M\) crosses the \(x\)-axis at the point \(N(n, 0)\).

1. Show that \(n = 14\). [1]

The point \(P(\ln 4, 13)\) lies on \(C\). The finite region \(R\) is bounded by \(C\), the axes and the line \(PN\), as shown in Fig. 3.

1. Find the area of \(R\), giving your answers in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found. [7]

The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_4}

1. Determine the coordinates of \(B\). [6]
2. Hence determine the exact value of \(\tan \alpha\). [3]

\includegraphics{figure_3} The curve \(C_1\), shown in Figure 3, has equation \(y = 4x^2 - 6x + 4\). The point \(P\left(\frac{1}{2}, 2\right)\) lies on \(C_1\) The curve \(C_2\), also shown in Figure 3, has equation \(y = \frac{1}{2}x + \ln(2x)\). The normal to \(C_1\) at the point \(P\) meets \(C_2\) at the point \(Q\). Find the exact coordinates of \(Q\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [8]

The diagram below shows part of a curve C with equation \(y = 1 + 3x - \frac{1}{2}x^2\). \includegraphics{figure_7}

1. The curve crosses the \(y\) axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the \(x\) coordinate of the point B. [4]
2. The region R is bounded by the curve C and the line L. Find the exact area of R. [3]

The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_7}

1. Determine the coordinates of \(B\). [6]
2. Hence determine the exact value of \(\tan\alpha\). [3]