Conic sections

2
\includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-2_728_951_486_534} The diagram shows the graph of $$y = \frac { 2 x ^ { 2 } + 3 x + 3 } { x + 1 }$$

1. Find the equations of the asymptotes of the curve.
2. Prove that the values of \(y\) between which there are no points on the curve are - 5 and 3 .

2 A curve has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Sketch the curve, showing the coordinates of the points of intersection with the coordinate axes.
2. Calculate the \(y\)-coordinates of the points of intersection of the curve with the line \(x = 1\). Give your answers in the form \(p \sqrt { 2 }\), where \(p\) is a rational number.
3. The curve is translated one unit in the positive \(x\) direction. Write down the equation of the curve after the translation.

6．（a）Given that \(x ^ { 4 } + y ^ { 4 } = 1\) ，prove that \(x ^ { 2 } + y ^ { 2 }\) is a maximum when \(x = \pm y\) ，and find the maximum and minimum values of \(x ^ { 2 } + y ^ { 2 }\) ．
（b）On the same diagram，sketch the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations \(x ^ { 4 } + y ^ { 4 } = 1\) and \(x ^ { 2 } + y ^ { 2 } = 1\) respectively．
（c）Write down the equation of the circle \(C _ { 3 }\) ，centre the origin，which touches the curve \(C _ { 1 }\) at the points where \(x = \pm y\) ．

1. The hyperbola \(H\) has foci at \(( 5,0 )\) and \(( - 5,0 )\) and directrices with equations \(x = \frac { 9 } { 5 }\) and \(x = - \frac { 9 } { 5 }\).

Find a cartesian equation for \(H\).

3 Find the equations of the asymptotes of the curve \(x ^ { 2 } - 3 y ^ { 2 } = 1\) Circle your answer.
[0pt] [1 mark] $$y = \pm 3 x \quad y = \pm \frac { 1 } { 3 } x \quad y = \pm \sqrt { 3 } x \quad y = \pm \frac { 1 } { \sqrt { 3 } } x$$ Turn over for the next question
\(\mathbf { 4 } \quad \mathbf { A } = \left[ \begin{array} { l l } 1 & 2 \\ 1 & k \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { c c } - 1 & 0 \\ 0 & 1 \end{array} \right]\)

7. A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1$$ where \(a\) is a positive constant.
The eccentricity of \(H\) is \(e\).

1. Determine an expression for \(e ^ { 2 }\) in terms of \(a\). The line \(l\) is the directrix of \(H\) for which \(x > 0\)
   The points \(A\) and \(A ^ { \prime }\) are the points of intersection of \(l\) with the asymptotes of \(H\).
2. Determine, in terms of \(e\), the length of the line segment \(A A ^ { \prime }\). The point \(F\) is the focus of \(H\) for which \(x < 0\)
   Given that the area of triangle \(A F A ^ { \prime }\) is \(\frac { 164 } { 3 }\)
3. show that \(a\) is a solution of the equation $$30 a ^ { 3 } - 164 a ^ { 2 } + 375 a - 4100 = 0$$
4. Hence, using algebra and making your reasoning clear, show that the only possible value of \(a\) is \(\frac { 20 } { 3 }\)

1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\)

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
Given that \(L\) is a tangent to \(E\),

1. show that $$c ^ { 2 } - 25 m ^ { 2 } = 9$$
2. find the equations of the tangents to \(E\) which pass through the point \(( 3,4 )\).

8 A curve \(C\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$

1. Find the \(y\)-coordinates of the points on \(C\) for which \(x = 10\), giving each answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
2. Sketch the curve \(C\), indicating the coordinates of any points where the curve intersects the coordinate axes.
3. Write down the equation of the tangent to \(C\) at the point where \(C\) intersects the positive \(x\)-axis.
4. 1. Show that, if the line \(y = x - 4\) intersects \(C\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$16 x ^ { 2 } - 200 x + 625 = 0$$
   2. Solve this equation and hence state the relationship between the line \(y = x - 4\) and the curve \(C\).

1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\)

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
Given that \(L\) is a tangent to \(E\),

1. show that $$c ^ { 2 } - 25 m ^ { 2 } = 9$$
2. find the equations of the tangents to \(E\) which pass through the point \(( 3,4 )\).

7. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a constant. The point \(P \left( c t , \frac { c } { t } \right)\) is a general point on \(H\).

1. Show that the tangent to \(H\) at \(P\) has equation $$t ^ { 2 } y + x = 2 c t$$ The tangents to \(H\) at the points \(A\) and \(B\) meet at the point \(( 15 c , - c )\).
2. Find, in terms of \(c\), the coordinates of \(A\) and \(B\).

7. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a constant. The point \(P \left( c t , \frac { c } { t } \right)\) is a general point on \(H\).

1. Show that the tangent to \(H\) at \(P\) has equation $$t ^ { 2 } y + x = 2 c t$$ The tangents to \(H\) at the points \(A\) and \(B\) meet at the point \(( 15 c , - c )\).
2. Find, in terms of \(c\), the coordinates of \(A\) and \(B\).

1. The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) lies on the parabola with equation \(y ^ { 2 } = 8 x\)
   1. Show that the point \(Q \left( \frac { 2 } { p ^ { 2 } } , \frac { - 4 } { p } \right)\), where \(p \neq 0\), lies on the parabola.
   2. Show that the chord \(P Q\) passes through the focus of the parabola.
   The tangent to the parabola at \(P\) and the tangent to the parabola at \(Q\) meet at the point \(R\)
2. Determine, in simplest form, the coordinates of \(R\)

4. The hyperbola \(H\) has equation $$x y = 3$$ The point \(Q ( 1,3 )\) is on \(H\).

1. Find the equation of the normal to \(H\) at \(Q\) in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
   (5) The normal at \(Q\) intersects \(H\) again at the point \(R\).
2. Find the coordinates of \(R\).
   (5)

1. The rectangular hyperbola \(H\) has equation \(x y = 25\)
   1. Verify that, for \(t \neq 0\), the point \(P \left( 5 t , \frac { 5 } { t } \right)\) is a general point on \(H\).
   The point \(A\) on \(H\) has parameter \(t = \frac { 1 } { 2 }\)
2. Show that the normal to \(H\) at the point \(A\) has equation $$8 y - 2 x - 75 = 0$$ This normal at \(A\) meets \(H\) again at the point \(B\).
3. Find the coordinates of \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-13_2261_50_312_36}

   VIAV SIHI NI BIIIM ION OC

   VGHV SIHI NI GHIYM ION OC

   VJ4V SIHI NI JIIYM ION OC

6. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a non-zero constant. The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).

1. Show that the normal to \(H\) at \(P\) has equation $$y p - p ^ { 3 } x = c \left( 1 - p ^ { 4 } \right)$$ The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
2. Find, in terms of \(c\) and \(p\), the coordinates of \(Q\).

2. A point \(P\) with coordinates \(( x , y )\) moves so that its distance from the point \(( - 3,0 )\) is equal to its distance from the line \(x = 3\). Find a cartesian equation for the locus of \(P\).

2. A point \(P\) with coordinates \(( x , y )\) moves so that its distance from the point \(( - 3,0 )\) is equal to its distance from the line \(x = 3\). Find a cartesian equation for the locus of \(P\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 10 x\)

The point \(F\) is the focus of \(C\).

1. Write down the coordinates of \(F\). The point \(P\) on \(C\) has \(y\) coordinate \(q\), where \(q > 0\)
2. Show that an equation for the tangent to \(C\) at \(P\) is given by $$10 x - 2 q y + q ^ { 2 } = 0$$ The tangent to \(C\) at \(P\) intersects the directrix of \(C\) at the point \(A\).
   The point \(B\) lies on the directrix such that \(P B\) is parallel to the \(x\)-axis.
3. Show that the point of intersection of the diagonals of quadrilateral \(P B A F\) always lies on the \(y\)-axis.

1. The parabola \(C\) has equation

$$y ^ { 2 } = 16 x$$ The distinct points \(P \left( p ^ { 2 } , 4 p \right)\) and \(Q \left( q ^ { 2 } , 4 q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0\)
The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) meet at the point \(R ( - 28,6 )\).
Show that the area of triangle \(P Q R\) is 1331

1. The hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\).
The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).

1. Determine, in terms of \(c\) and \(t\),
   1. the coordinates of \(A\),
   2. the coordinates of \(B\). Given that the area of triangle \(A O B\), where \(O\) is the origin, is 90 square units,
2. determine the value of \(c\), giving your answer as a simplified surd.

1. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x\) where \(a\) is a positive constant.

The point \(S\) is the focus of \(C\)
The line \(l _ { 1 }\) with equation \(y = k\) where \(k\) is a positive constant, intersects \(C\) at the point \(P\)

1. Show that $$P S = \frac { k ^ { 2 } + 4 a ^ { 2 } } { 4 a }$$ The line \(l _ { 2 }\) passes through \(P\) and intersects the directrix of \(C\) on the \(x\)-axis.
   The line \(l _ { 2 }\) intersects the \(y\)-axis at the point \(A\)
2. Show that the \(y\) coordinate of \(A\) is \(\frac { 4 a ^ { 2 } k } { k ^ { 2 } + 4 a ^ { 2 } }\) The line \(l _ { 1 }\) intersects the directrix of \(C\) at the point \(B\)
   Given that the areas of triangles \(B P A\) and \(O S P\), where \(O\) is the origin, satisfy the ratio $$\text { area } B P A \text { : area } O S P = 4 k ^ { 2 } : 1$$
3. determine the exact value of \(a\)

1. \(P\) and \(Q\) are two distinct points on the ellipse described by the equation \(x ^ { 2 } + 4 y ^ { 2 } = 4\)

The line \(l\) passes through the point \(P\) and the point \(Q\).
The tangent to the ellipse at \(P\) and the tangent to the ellipse at \(Q\) intersect at the point \(( r , s )\).
Show that an equation of the line \(l\) is $$4 s y + r x = 4$$

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(P \left( a p ^ { 2 } \right.\), 2ap) lies on \(C\) where \(p > 0\)

1. Write down the coordinates of \(S\).
2. Write down the length of SP in terms of \(a\) and \(p\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p \neq q\), also lies on \(C\).
   The point \(M\) is the midpoint of \(P Q\).
   Given that \(p q = - 1\)
3. prove that, as \(P\) varies, the locus of \(M\) has equation $$y ^ { 2 } = 2 a ( x - a )$$

3. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 8 \cos \theta , 6 \sin \theta )\).

1. Using calculus, show that an equation for \(l\) is $$4 x \sin \theta - 3 y \cos \theta = 14 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
   The point \(M\) is the midpoint of \(A B\).
2. Determine a Cartesian equation for the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(a x ^ { 2 } + b y ^ { 2 } = c\) where \(a , b\) and \(c\) are integers.

7 Given that \(x = 2 \sec \theta\) and \(y = 3 \tan \theta\), show that \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1\).

8. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 36 x\). The point \(P \left( 9 p ^ { 2 } , 18 p \right)\), where \(p\) is a positive constant, lies on \(C\).

1. Using calculus, show that an equation of the tangent to \(C\) at \(P\) is $$p y - x = 9 p ^ { 2 }$$ This tangent cuts the directrix of \(C\) at the point \(A ( - a , 6 )\), where \(a\) is a constant.
2. Write down the value of \(a\).
3. Find the exact value of \(p\).
4. Hence find the exact coordinates of the point \(P\), giving each coordinate as a simplified surd.

2. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 25 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 6 \cos \theta , 5 \sin \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)

1. Use calculus to show that an equation of \(l\) is $$6 x \sin \theta - 5 y \cos \theta = 11 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(Q\). The point \(R\) is the foot of the perpendicular from \(P\) to the \(x\)-axis.
2. Show that \(\frac { O Q } { O R } = e ^ { 2 }\), where \(e\) is the eccentricity of the ellipse \(E\).

9. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)

1. Show that an equation of the normal to \(C\) at \(P ( a \sec \theta , b \tan \theta )\) is $$b y + a x \sin \theta = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(A B\) is \(M\).
2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies.
   (Total 13 marks)

2. The line with equation \(x = 9\) is a directrix of an ellipse with equation $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 8 } = 1$$ where \(a\) is a positive constant. Find the two possible exact values of the constant \(a\).

3. A parabola \(C\) has cartesian equation \(y ^ { 2 } = 16 x\). The point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).

1. Write down the coordinates of the focus \(F\) and the equation of the directrix of \(C\).
2. Show that the equation of the normal to \(C\) at \(P\) is \(y + t x = 8 t + 4 t ^ { 3 }\).

4. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 12 x\) The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on \(C\), where \(p \neq 0\)

1. Show that the equation of the normal to the curve \(C\) at the point \(P\) is $$y + p x = 6 p + 3 p ^ { 3 }$$ This normal crosses the curve \(C\) again at the point \(Q\).
   Given that \(p = 2\) and that \(S\) is the focus of the parabola, find
2. the coordinates of the point \(Q\),
3. the area of the triangle \(P Q S\).

8. The parabola \(C\) has equation \(y ^ { 2 } = 48 x\). The point \(P \left( 12 t ^ { 2 } , 24 t \right)\) is a general point on \(C\).

1. Find the equation of the directrix of \(C\).
2. Show that the equation of the tangent to \(C\) at \(P \left( 12 t ^ { 2 } , 24 t \right)\) is $$x - t y + 12 t ^ { 2 } = 0$$ The tangent to \(C\) at the point \(( 3,12 )\) meets the directrix of \(C\) at the point \(X\).
3. Find the coordinates of \(X\).

16. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\).

1. Show that an equation of the normal to \(C\) at the point \(P ( a \sec t , b \tan t )\) is $$a x \sin t + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan t .$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and \(S\) is a focus of \(C\). Given that the eccentricity of \(C\) is \(\frac { 3 } { 2 }\), and that \(O A = 3 O S\), where \(O\) is the origin,
2. determine the possible values of \(t\), for \(0 \leq t < 2 \pi\).
   [0pt] [P5 June 2003 Qn 1]

6. The rectangular hyperbola, \(H\), has cartesian equation $$x y = 36$$ The three points \(P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)\) and \(R \left( 6 r , \frac { 6 } { r } \right)\), where \(p , q\) and \(r\) are distinct, non-zero values, lie on the hyperbola \(H\).

1. Show that an equation of the line \(P Q\) is $$p q y + x = 6 ( p + q )$$ Given that \(P R\) is perpendicular to \(Q R\),
2. show that the normal to the curve \(H\) at the point \(R\) is parallel to the line \(P Q\).

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The line \(l\) has equation \(x - 2 y = c\)
The points \(P\) and \(Q\) are the points of intersection of \(H\) and \(l\)

1. Determine, in terms of \(c\), the coordinates of \(P\) and the coordinates of \(Q\) The point \(R\) is the midpoint of \(P Q\)
2. Show that, as \(C\) varies, the coordinates of \(R\) satisfy the equation $$x y = - \frac { c ^ { 2 } } { a }$$ where \(a\) is a constant to be determined.

8

1. The line \(y = m x\) is a tangent to \(P _ { 2 }\)
   Prove that \(m = \pm \sqrt { \frac { a } { b } }\)
   Solutions using differentiation will be given no marks.
   8
2. The line \(y = \sqrt { \frac { a } { b } } x\) meets \(P _ { 2 }\) at the point \(D\).
   The finite region \(R\) is bounded by the \(x\)-axis, \(P _ { 2 }\) and a line through \(D\) perpendicular to the \(x\)-axis. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
   Find, in terms of \(a\) and \(b\), the volume of this solid.
   Fully justify your answer.
3. Find the eigenvalues and corresponding eigenvectors of the matrix

6. The rectangular hyperbola \(H\) has cartesian equation \(x y = c ^ { 2 }\). The point \(P \left( c t , \frac { c } { t } \right) , t > 0\), is a general point on \(H\).

1. Show that an equation of the tangent to \(H\) at the point \(P\) is $$t ^ { 2 } y + x = 2 c t$$ An equation of the normal to \(H\) at the point \(P\) is \(t ^ { 3 } x - t y = c t ^ { 4 } - c\) Given that the normal to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and the tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(B\),
2. find, in terms of \(c\) and \(t\), the coordinates of \(A\) and the coordinates of \(B\). Given that \(c = 4\),
3. find, in terms of \(t\), the area of the triangle \(A P B\). Give your answer in its simplest form.