Parabola tangent intersection problems

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 tangent to \(C\) at \(P\) is $$p y = x + a p ^ { 2 }$$ The tangent to \(C\) at the point \(P\) intersects the directrix of \(C\) at the point \(B\) and intersects the \(x\)-axis at the point \(D\). Given that the \(y\)-coordinate of \(B\) is \(\frac { 5 } { 6 } a\) and \(p > 0\),
2. find, in terms of \(a\), the \(x\)-coordinate of \(D\). Given that \(O\) is the origin,
3. find, in terms of \(a\), the area of the triangle \(O P D\), giving your answer in its simplest form.

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.

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 $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$

8. A parabola has equation \(y ^ { 2 } = 4 a x , a > 0\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\) lies on the parabola.

1. Show that an equation of the tangent to the parabola at \(Q\) is $$y q = x + a q ^ { 2 }$$ This tangent meets the \(y\)-axis at the point \(R\).
2. Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\).
3. Show that \(l\) passes through the focus of the parabola.
4. Find the coordinates of the point where \(I\) meets the directrix of the parabola.

8. A parabola has equation \(y ^ { 2 } = 4 a x , a > 0\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\) lies on the parabola.

1. Show that an equation of the tangent to the parabola at \(Q\) is $$y q = x + a q ^ { 2 } .$$ This tangent meets the \(y\)-axis at the point \(R\).
2. Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\).
3. Show that \(l\) passes through the focus of the parabola.
4. Find the coordinates of the point where \(l\) meets the directrix of the parabola.

7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant and \(a > 0\) The point \(Q \left( a q ^ { 2 } , 2 a q \right) , q > 0\), lies on the parabola \(C\).

1. Show that an equation of the tangent to \(C\) at \(Q\) is $$q y = x + a q ^ { 2 }$$ The tangent to \(C\) at the point \(Q\) meets the \(x\)-axis at the point \(X \left( - \frac { 1 } { 4 } a , 0 \right)\) and meets the directrix of \(C\) at the point \(D\).
2. Find, in terms of \(a\), the coordinates of \(D\). Given that the point \(F\) is the focus of the parabola \(C\),
3. find the area, in terms of \(a\), of the triangle \(F X D\), giving your answer in its simplest form.

A parabola has equation \(y^2 = 4ax\), \(a > 0\). The point \(Q (aq^2, 2aq)\) lies on the parabola.

1. Show that an equation of the tangent to the parabola at \(Q\) is $$yq = x + aq^2.$$ [4]
2. This tangent meets the \(y\)-axis at the point \(R\). Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\). [3]
3. Show that \(l\) passes through the focus of the parabola. [1]
4. Find the coordinates of the point where \(l\) meets the directrix of the parabola. [2]

The parabola \(C\) has equation \(y^2 = 4ax\), where \(a\) is a positive constant. The point \(P(at^2, 2at)\) is a general point on \(C\).

1. Show that the equation of the tangent to \(C\) at \(P(at^2, 2at)\) is $$ty = x + at^2$$ [4]

The tangent to \(C\) at \(P\) meets the \(y\)-axis at a point \(Q\).

1. Find the coordinates of \(Q\). [1]

Given that the point \(S\) is the focus of \(C\),

1. show that \(PQ\) is perpendicular to \(SQ\). [3]