1.07m Tangents and normals: gradient and equations

1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1\)

The point \(P ( 4 \cos \theta , 3 \sin \theta )\) lies on \(E\).

1. Use calculus to show that an equation of the tangent to \(E\) at \(P\) is $$3 x \cos \theta + 4 y \sin \theta = 12$$
2. Determine an equation for the normal to \(E\) at \(P\). The tangent to \(E\) at \(P\) meets the \(x\)-axis at the point \(A\).
   The normal to \(E\) at \(P\) meets the \(y\)-axis at the point \(B\).
3. Show that the locus of the midpoint of \(A\) and \(B\) as \(\theta\) varies has equation $$x ^ { 2 } \left( p - q y ^ { 2 } \right) = r$$ where \(p , q\) and \(r\) are integers to be determined.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 5 \cos \theta , 3 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$5 x \sin \theta - 3 y \cos \theta = 16 \sin \theta \cos \theta$$ Given that
   • \(\quad l\) intersects the \(y\)-axis at the point \(Q\)
   • the midpoint of the line segment \(P Q\) is \(M\)
   • determine the exact maximum area of triangle \(O M P\) as \(\theta\) varies, where \(O\) is the origin.
   You must justify your answer.

8. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\). The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).

1. Use calculus to show that an equation of \(l _ { 1 }\) is $$2 y \sin t = x - 4 \cos t$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\).
2. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line \(l _ { 1 }\) is a tangent to \(E\) at the point \(P ( a \cos \theta , b \sin \theta )\).

1. Using calculus, show that an equation for \(l _ { 1 }\) is $$\frac { x \cos \theta } { a } + \frac { y \sin \theta } { b } = 1$$ The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$$ The line \(l _ { 2 }\) is a tangent to \(C\) at the point \(Q ( a \cos \theta , a \sin \theta )\).
2. Find an equation for the line \(l _ { 2 }\). Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\),
3. find, in terms of \(a , b\) and \(\theta\), the coordinates of \(R\).
4. Find the locus of \(R\), as \(\theta\) varies.
   

5

1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7 .$$
2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

9

1. Find the gradient of the curve \(y = 2 x ^ { 2 }\) at the point where \(x = 3\).
2. At a point \(A\) on the curve \(y = 2 x ^ { 2 }\), the gradient of the normal is \(\frac { 1 } { 8 }\). Find the coordinates of \(A\). Points \(P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)\) and \(P _ { 3 } \left( 1.1 , y _ { 3 } \right)\) lie on the curve \(y = k x ^ { 2 }\). The gradient of the chord \(P _ { 1 } P _ { 3 }\) is 6.3 and the gradient of the chord \(P _ { 1 } P _ { 2 }\) is 6.03.
3. What do these results suggest about the gradient of the tangent to the curve \(y = k x ^ { 2 }\) at \(P _ { 1 }\) ?
4. Deduce the value of \(k\).

8

1. Given that \(y = x ^ { 2 } - 5 x + 15\) and \(5 x - y = 10\), show that \(x ^ { 2 } - 10 x + 25 = 0\).
2. Find the discriminant of \(x ^ { 2 } - 10 x + 25\).
3. What can you deduce from the answer to part (ii) about the line \(5 x - y = 10\) and the curve \(y = x ^ { 2 } - 5 x + 15\) ?
4. Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 15 \text { and } 5 x - y = 10$$
5. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 5 x + 15\) at the point \(( 5,15 )\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.

10

1. Given that \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the coordinates of the stationary points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\).
3. Determine whether each stationary point is a maximum point or a minimum point.
4. Given that \(24 x + 3 y + 2 = 0\) is the equation of the tangent to the curve at the point ( \(p , q\) ), find \(p\) and \(q\).

10

1. Solve the equation \(3 x ^ { 2 } - 14 x - 5 = 0\). A curve has equation \(\mathrm { y } = 3 \mathrm { x } ^ { 2 } - 14 \mathrm { x } - 5\).
2. Sketch the curve, indicating the coordinates of all intercepts with the axes.
3. Find the value of C for which the line \(\mathrm { y } = 4 \mathrm { x } + \mathrm { C }\) is a tangent to the curve.

9

1. Find the equation of the circle with radius 10 and centre ( 2,1 ), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. The circle passes through the point \(( 5 , k )\) where \(k > 0\). Find the value of \(k\) in the form \(p + \sqrt { q }\).
3. Determine, showing all working, whether the point \(( - 3,9 )\) lies inside or outside the circle.
4. Find an equation of the tangent to the circle at the point ( 8,9 ).

5

1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7$$
2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

11 \begin{figure}[h] \end{figure} A circle has centre \(C ( 1,3 )\) and passes through the point \(A ( 3,7 )\) as shown in Fig. 11.

1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.

9. A curve has the equation \(y = \frac { x } { 2 } + 3 - \frac { 1 } { x } , x \neq 0\). The point \(A\) on the curve has \(x\)-coordinate 2 .

1. Find the gradient of the curve at \(A\).
2. Show that the tangent to the curve at \(A\) has equation $$3 x - 4 y + 8 = 0$$ The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
3. Find the coordinates of \(B\).

9. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).

1. Find an equation for \(C\).
2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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.

9.

1. Find an equation for the tangent to the curve \(y = x ^ { 2 } + 2\) at the point \(( 1,3 )\) in the form \(y = m x + c\).
2. Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers.
3. Describe fully the transformation that maps the graph of \(y = x ^ { 2 } + 2\) onto the graph of \(y = x ^ { 2 } - 6 x + 11\).
4. Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve \(y = x ^ { 2 } - 6 x + 11\) at the point with \(x\)-coordinate 4.

10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 2 ) ^ { 3 }$$

1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
4. Show that \(m\) has the equation \(y = 3 x + 8\).

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.

1. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2 .

Find an equation for the tangent to the curve at \(A\).

10. The curve with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).

1. Sketch the curve, showing the coordinates of \(A\) and \(B\).
2. Show that the tangent to the curve at \(A\) has the equation $$x + y = 2$$ Given that the curve is stationary at the points \(B\) and \(C\),
3. find the exact coordinates of \(C\).

8.

1. Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
2. Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
3. Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).

10. \includegraphics[max width=\textwidth, alt={}, center]{76efaf91-a6f3-4493-88d4-3654b023441d-3_646_773_986_477} The diagram shows the curve \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = 2 x + 1\). The curve and line intersect at the points \(P\) and \(Q\).

1. Using algebra, show that \(P\) has coordinates \(( 1,3 )\) and find the coordinates of \(Q\).
2. Find an equation for the tangent to the curve at \(P\).
3. Show that the tangent to the curve at \(Q\) has the equation \(y = 5 x - 11\).
4. Find the coordinates of the point where the tangent to the curve at \(P\) intersects the tangent to the curve at \(Q\).

10 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).

1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).

12 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h] \end{figure}

1. Show that the curve intersects the \(x\)-axis at \(( 4,0 )\) and find the coordinates of the other point of intersection of the curve and the \(x\)-axis.
2. Find the equation of the normal to the curve at the point \(( 4,0 )\). Show also that the area of the triangle bounded by this normal and the axes is 1.6 units \(^ { 2 }\).
3. Find the area of the region bounded by the curve and the \(x\)-axis.

9 \begin{figure}[h] \end{figure} Fig. 9 shows a sketch of the graph of \(y = x ^ { 3 } - 10 x ^ { 2 } + 12 x + 72\).

1. Write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\).
3. Show that the curve crosses the \(x\)-axis at \(x = - 2\). Show also that the curve touches the \(x\)-axis at \(x = 6\).
4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9 . [4]