Normal meets curve/axis — further geometry

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation $$y = \frac { 2 } { 7 } x ^ { 3 } + \frac { 1 } { 7 } x ^ { 2 } - \frac { 5 } { 2 } x + k$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The line \(l\), shown in Figure 5, is the normal to \(C\) at the point \(A\) with \(x\) coordinate \(- \frac { 7 } { 2 }\) Given that \(l\) is also a tangent to \(C\) at the point \(B\),
2. show that the \(x\) coordinate of the point \(B\) is a solution of the equation $$12 x ^ { 2 } + 4 x - 33 = 0$$
3. Hence find the \(x\) coordinate of \(B\), justifying your answer. Given that the \(y\) intercept of \(l\) is - 1
4. find the value of \(k\).
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-32_2640_1840_118_114}

13. The curve \(C\) has equation $$y = 3 x ^ { 2 } - 4 x + 2$$ The line \(l _ { 1 }\) is the normal to the curve \(C\) at the point \(P ( 1,1 )\)

1. Show that \(l _ { 1 }\) has equation $$x + 2 y - 3 = 0$$ The line \(l _ { 1 }\) meets curve \(C\) again at the point \(Q\).
2. By solving simultaneous equations, determine the coordinates of the point \(Q\). Another line \(l _ { 2 }\) has equation \(k x + 2 y - 3 = 0\), where \(k\) is a constant.
3. Show that the line \(l _ { 2 }\) meets the curve \(C\) once only when $$k ^ { 2 } - 16 k + 40 = 0$$
4. Find the two exact values of \(k\) for which \(l _ { 2 }\) is a tangent to \(C\).

8. The curve \(C\) has equation \(y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the point \(P ( 4,8 )\) lies on \(C\).
3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3 y = x + 20 .$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).
4. Find the length \(P Q\), giving your answer in a simplified surd form.

1. The curve \(C\) has equation

$$y = 9 - 4 x - \frac { 8 } { x } , \quad x > 0$$ The point \(P\) on \(C\) has \(x\)-coordinate equal to 2 .

1. Show that the equation of the tangent to \(C\) at the point \(P\) is \(y = 1 - 2 x\).
2. Find an equation of the normal to \(C\) at the point \(P\). The tangent at \(P\) meets the \(x\)-axis at \(A\) and the normal at \(P\) meets the \(x\)-axis at \(B\).
3. Find the area of triangle \(A P B\).

11. \begin{figure}[h] \end{figure} A sketch of part of the curve \(C\) with equation $$y = 20 - 4 x - \frac { 18 } { x } , \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has an \(x\) coordinate equal to 2

1. Show that the equation of the normal to \(C\) at \(A\) is \(y = - 2 x + 7\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3 .
2. Use algebra to find the coordinates of \(B\).

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x + \frac { 27 } { x } - 12 , \quad x > 0$$ The point \(A\) lies on \(C\) and has coordinates \(\left( 3 , - \frac { 3 } { 2 } \right)\).

1. Show that the equation of the normal to \(C\) at \(A\) can be written as \(10 y = 4 x - 27\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.
2. Use algebra to find the coordinates of \(B\).

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
3. Show that the normal to the curve at \(P\) does not intersect the curve again.

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

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

9. \includegraphics[max width=\textwidth, alt={}, center]{33f9663f-26bb-445e-af6e-ca5ca927f7dd-3_638_757_1064_493} The diagram 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 \(y = m x + c\).
2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again.
3. Show that the area of the shaded region bounded by the curve and the straight line \(P Q\) is \(\frac { 4 } { 3 }\).

5. A curve has the equation \(y = \sqrt { 3 x + 11 }\). The point \(P\) on the curve has \(x\)-coordinate 3 .

1. Show that the tangent to the curve at \(P\) has the equation $$3 x - 4 \sqrt { 5 } y + 31 = 0$$ The normal to the curve at \(P\) crosses the \(y\)-axis at \(Q\).
2. Find the \(y\)-coordinate of \(Q\) in the form \(k \sqrt { 5 }\).

10 A curve has equation \(y = x ^ { 2 } + x\).

1. Find the gradient of the curve at the point for which \(x = 2\).
2. Find the equation of the normal to the curve at the point for which \(x = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find the values of \(k\) for which the line \(y = k x - 4\) is a tangent to the curve.

11 The point \(P\) on the curve \(y = k \sqrt { x }\) has \(x\)-coordinate 4 . The normal to the curve at \(P\) is parallel to the line \(2 x + 3 y = 0\).

1. Find the value of \(k\).
2. This normal meets the \(x\)-axis at the point \(Q\). Calculate the area of the triangle \(O P Q\), where \(O\) is the point \(( 0,0 )\). RECOGNISING ACHIEVEMENT

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

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

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

1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).

5 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(P ( 4,0 )\).
The normal to the curve at \(P\) meets the \(y\)-axis at the point \(Q\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-3_526_629_916_813} The curve, defined for \(x \geqslant 0\), has equation $$y = 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } }$$

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (3 marks)
   2. Show that the gradient of the curve at \(P ( 4,0 )\) is - 2 .
   3. Find an equation of the normal to the curve at \(P ( 4,0 )\).
   4. Find the \(y\)-coordinate of \(Q\) and hence find the area of triangle \(O P Q\).
   5. The curve has a maximum point \(M\). Find the \(x\)-coordinate of \(M\).
2. 1. Find \(\int \left( 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
   2. Find the total area of the region bounded by the curve and the lines \(P Q\) and \(Q O\).

5 The point \(P ( 2,8 )\) lies on a curve, and the point \(M\) is the only stationary point of the curve. The curve has equation \(y = 6 + 2 x - \frac { 8 } { x ^ { 2 } }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the normal to the curve at the point \(P ( 2,8 )\) has equation \(x + 4 y = 34\).
3. 1. Show that the stationary point \(M\) lies on the \(x\)-axis.
   2. Hence write down the equation of the tangent to the curve at \(M\).
4. The tangent to the curve at \(M\) and the normal to the curve at \(P\) intersect at the point \(T\). Find the coordinates of \(T\).

2. A curve has the equation \(y = \sqrt { 3 x + 11 }\). The point \(P\) on the curve has \(x\)-coordinate 3 .

1. Show that the tangent to the curve at \(P\) has the equation $$3 x - 4 \sqrt { 5 } y + 31 = 0$$ The normal to the curve at \(P\) crosses the \(y\)-axis at \(Q\).
2. Find the \(y\)-coordinate of \(Q\) in the form \(k \sqrt { 5 }\).

8 A curve has equation $$y = x ^ { 3 } + p x ^ { 2 } + q x - 45$$ The curve passes through point \(R ( 2,3 )\) The gradient of the curve at \(R\) is 8
8

1. Find the value of \(p\) and the value of \(q\).
   8
2. Calculate the area enclosed between the normal to the curve at \(R\) and the coordinate 8 (b) axes. \(9 \quad\) A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ( x - 3 ) ^ { 2 }$$

4

1. Sketch the graph of \(y = \sqrt { 2 } \sin x\) for \(0 \leqslant x \leqslant 2 \pi\). The points \(P\) and \(Q\) on the graph have \(x\)-coordinates \(\frac { 1 } { 4 } \pi\) and \(\frac { 3 } { 4 } \pi\), respectively.
2. Determine the equation of the tangent to the curve at \(P\). The normals to the curve at \(P\) and \(Q\) intersect at the point \(R\).
3. Determine the exact coordinates of \(R\).

\includegraphics{figure_3} A sketch of part of the curve \(C\) with equation $$y = 20 - 4x - \frac{18}{x}, \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has \(x\) coordinate equal to 2

1. Show that the equation of the normal to \(C\) at \(A\) is \(y = -2x + 7\). [6]

The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.

1. Use algebra to find the coordinates of \(B\). [5]

The curve \(C\) has equation \(y = 4x + 3x^{-1} - 2x^2\), \(x > 0\).

1. Find an expression for \(\frac{dy}{dx}\). [3]
2. Show that the point \(P(4, 8)\) lies on \(C\). [1]
3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3y - x + 20.$$ [4]

The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).

1. Find the length \(PQ\), giving your answer in a simplified surd form. [3]

The curve \(C\) has equation \(y = x^3 - 5x + \frac{2}{x}\), \(x \neq 0\). The points \(A\) and \(B\) both lie on \(C\) and have coordinates \((1, -2)\) and \((-1, 2)\) respectively.

1. Show that the gradient of \(C\) at \(A\) is equal to the gradient of \(C\) at \(B\). [5]
2. Show that an equation for the normal to \(C\) at \(A\) is \(4y = x - 9\). [4]

The normal to \(C\) at \(A\) meets the \(y\)-axis at the point \(P\). The normal to \(C\) at \(B\) meets the \(y\)-axis at the point \(Q\).

1. Find the length of \(PQ\). [4]

The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P(\ln 2, p + 2q)\) on \(C\), the gradient is \(5\).

1. Find the value of \(p\) and the value of \(q\). [5]

The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).

1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately \(53.8\). [5]

The curve \(C\) has equation \(y = f(x)\), where $$f(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$ 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 \(y = x + \frac{3}{x}\), \(x \neq 0\). 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 \(y = mx + c\). [4]
3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again. [4]