1.07m Tangents and normals: gradient and equations

1. The parabola \(C\) has equation \(y ^ { 2 } = \frac { 4 } { 3 } x\)

The point \(P \left( \frac { 1 } { 3 } t ^ { 2 } , \frac { 2 } { 3 } t \right)\), where \(t \neq 0\), lies on \(C\).

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$3 t x + 3 y = t ^ { 3 } + 2 t$$ The normal to \(C\) at the point where \(t = 9\) meets \(C\) again at the point \(Q\).
2. Determine the exact coordinates of \(Q\).

8. The hyperbola \(H\) has cartesian equation \(x y = 16\) The parabola \(P\) has parametric equations \(x = 8 t ^ { 2 } , y = 16 t\).

1. Find, using algebra, the coordinates of the point \(A\) where \(H\) meets \(P\). Another point \(B ( 8,2 )\) lies on the hyperbola \(H\).
2. Find the equation of the normal to \(H\) at the point (8, 2), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
3. Find the coordinates of the points where this normal at \(B\) meets the parabola \(P\).

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 \(P Q\).

6. 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 \(B\).
   

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.

10. The rectangular hyperbola \(H\) has equation \(x y = 144\). The point \(P\), on \(H\), has coordinates \(\left( 12 p , \frac { 12 } { p } \right)\), where \(p\) is a non-zero constant.

1. Show, by using calculus, that the normal to \(H\) at the point \(P\) has equation $$y = p ^ { 2 } x + \frac { 12 } { p } - 12 p ^ { 3 }$$ Given that the normal through \(P\) crosses the positive \(x\)-axis at the point \(Q\) and the negative \(y\)-axis at the point \(R\),
2. find the coordinates of \(Q\) and the coordinates of \(R\), giving your answers in terms of \(p\).
3. Given also that the area of triangle \(O Q R\) is 512 , find the possible values of \(p\).
   

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1. The rectangular hyperbola \(H\) has equation \(x y = 64\)

The point \(P \left( 8 p , \frac { 8 } { p } \right)\), where \(p \neq 0\), lies on \(H\).

1. Use calculus to show that the normal to \(H\) at \(P\) has equation $$p ^ { 3 } x - p y = 8 \left( p ^ { 4 } - 1 \right)$$ The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
2. Determine, in terms of \(p\), the coordinates of \(Q\), giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-17_2255_50_314_34}
   

7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The line \(l\) with equation \(3 x - 4 y + 48 = 0\) is a tangent to \(C\) at the point \(P\).

1. Show that \(a = 9\)
2. Hence determine the coordinates of \(P\). Given that the point \(S\) is the focus of \(C\) and that the line \(l\) crosses the directrix of \(C\) at the point \(A\),
3. determine the exact area of triangle \(P S A\). \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-25_2255_50_314_34}
   

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1. The rectangular 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 \(H\)

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\) The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
2. Determine the exact coordinates of \(Q\) Given that
   • the point \(R\) is the focus of \(C\)
   • the line \(l\) is the directrix of \(C\)
   • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
   • determine the exact length of \(Q S\)

6. The curve \(H\) has equation $$x y = a ^ { 2 } \quad x > 0$$ where \(a\) is a positive constant. The line with equation \(y = k x\), where \(k\) is a positive constant, intersects \(H\) at the point \(P\)

1. Use calculus to determine, in terms of \(a\) and \(k\), an equation for the tangent to \(H\) at \(P\) The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\)
2. Determine the coordinates of \(A\) and the coordinates of \(B\), giving your answers in terms of \(a\) and \(k\)
3. Hence show that the area of triangle \(A O B\), where \(O\) is the origin, is independent of \(k\)

1. The parabola \(C\) has equation \(y ^ { 2 } = 20 x\)

The point \(P\) on \(C\) has coordinates ( \(5 p ^ { 2 } , 10 p\) ) where \(p\) is a non-zero constant.

1. Use calculus to show that the tangent to \(C\) at \(P\) has equation $$p y - x = 5 p ^ { 2 }$$ The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
2. Write down the coordinates of \(A\). The point \(S\) is the focus of \(C\).
3. Write down the coordinates of \(S\). The straight line \(l _ { 1 }\) passes through \(A\) and \(S\).
   The straight line \(l _ { 2 }\) passes through \(O\) and \(P\), where \(O\) is the origin. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(B\),
4. show that the coordinates of \(B\) satisfy the equation $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$

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}

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1. The point \(\mathrm { P } \left( 6 \mathrm { t } , \frac { 6 } { \mathrm { t } } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\) (a) Show that an equation for the tangent to \(H\) at \(P\) is

$$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
(b) Find the coordinates of \(A\) and \(B\).

9. The rectangular hyperbola \(H\) has cartesian equation \(x y = 9\) The points \(P \left( 3 p , \frac { 3 } { p } \right)\) and \(Q \left( 3 q , \frac { 3 } { q } \right)\) lie on \(H\), where \(p \neq \pm q\).

1. Show that the equation of the tangent at \(P\) is \(x + p ^ { 2 } y = 6 p\).
2. Write down the equation of the tangent at \(Q\). The tangent at the point \(P\) and the tangent at the point \(Q\) intersect at \(R\).
3. Find, as single fractions in their simplest form, the coordinates of \(R\) in terms of \(p\) and \(q\).

7. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\) The point \(P \left( 5 p , \frac { 5 } { p } \right)\), and the point \(Q \left( 5 q , \frac { 5 } { q } \right)\), where \(p , q \neq 0 , p \neq q\), are points on the rectangular hyperbola \(H\).

1. Show that the equation of the tangent at point \(P\) is $$p ^ { 2 } y + x = 10 p$$
2. Write down the equation of the tangent at point \(Q\). The tangents at \(P\) and \(Q\) meet at the point \(N\).
   Given \(p + q \neq 0\),
3. show that point \(N\) has coordinates \(\left( \frac { 10 p q } { p + q } , \frac { 10 } { p + q } \right)\). The line joining \(N\) to the origin is perpendicular to the line \(P Q\).
4. Find the value of \(p ^ { 2 } q ^ { 2 }\).

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the parabola with equation \(y ^ { 2 } = 36 x\).
The point \(P ( 4,12 )\) lies on the parabola.

1. Find an equation for the normal to the parabola at \(P\). This normal meets the \(x\)-axis at the point \(N\) and \(S\) is the focus of the parabola, as shown in Figure 1.
2. Find the area of triangle \(P S N\).

8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a p ^ { 2 } , 2 a p \right)\) lies on the parabola \(C\).

1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = a p ^ { 3 } + 2 a p$$ The normal to \(C\) at the point \(P\) meets the \(x\)-axis at the point \(( 6 a , 0 )\) and meets the directrix of \(C\) at the point \(D\). Given that \(p > 0\),
2. find, in terms of \(a\), the coordinates of \(D\). Given also that the directrix of \(C\) cuts the \(x\)-axis at the point \(X\),
3. find, in terms of \(a\), the area of the triangle XPD, giving your answer in its simplest form.

6. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\).

1. Verify that the point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
2. Write down the coordinates of the focus \(S\) of \(C\).
3. Show that the normal to \(C\) at \(P\) has equation $$y + t x = 8 t + 4 t ^ { 3 }$$ The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(N\).
4. Find the area of triangle \(P S N\) in terms of \(t\), giving your answer in its simplest form.

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where c is a positive constant. The point \(A\) on \(H\) has \(x\)-coordinate \(3 c\).

1. Write down the \(y\)-coordinate of \(A\).
2. Show that an equation of the normal to \(H\) at \(A\) is $$3 y = 27 x - 80 c$$ The normal to \(H\) at \(A\) meets \(H\) again at the point \(B\).
3. Find, in terms of \(c\), the coordinates of \(B\).

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 \(X\).

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant. The point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\).

1. Show that an equation for the tangent to \(H\) at \(P\) is $$x + t ^ { 2 } y = 2 c t$$ The tangent to \(H\) at the point \(P\) meets the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Given that the area of the triangle \(O A B\), where \(O\) is the origin, is 36 ,
2. find the exact value of \(c\), expressing your answer in the form \(k \sqrt { } 2\), where \(k\) is an integer.

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)

6. A curve \(C\) is in the form of a parabola with equation \(y ^ { 2 } = 4 x\). \(P \left( p ^ { 2 } , 2 p \right)\) and \(Q \left( q ^ { 2 } , 2 q \right)\) are points on \(C\) where \(p > q\).

1. Find an equation of the tangent to \(C\) at \(P\).
   (5)
2. The tangent at \(P\) and the tangent at \(Q\) are perpendicular and intersect at the point \(R ( - 1,2 )\).
   1. Find the exact value of \(p\) and the exact value of \(q\).
   2. Find the area of the triangle \(P Q R\).

4. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 4\) The point \(P \left( 2 t , \frac { 2 } { t } \right)\) lies on \(H\), where \(t \neq 0\)

1. Show that an equation of the normal to \(H\) at the point \(P\) is $$t y - t ^ { 3 } x = 2 - 2 t ^ { 4 }$$ The normal to \(H\) at the point where \(t = - \frac { 1 } { 2 }\) meets \(H\) again at the point \(Q\).
2. Find the coordinates of the point \(Q\).

6. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x , \quad a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0 , p \neq q\).

1. Show that an equation of the tangent to the parabola at \(P\) is $$p y - x = a p ^ { 2 }$$
2. Write down the equation of the tangent at \(Q\). The tangent at \(P\) meets the tangent at \(Q\) at the point \(R\).
3. Find, in terms of \(p\) and \(q\), the coordinates of \(R\), giving your answers in their simplest form. Given that \(R\) lies on the directrix of \(C\),
4. find the value of \(p q\).