Conic tangent through external point

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 \(( 3,4 )\).

5. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1\) The line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants. Given that \(l\) is a tangent to \(H\),

1. show that \(25 m ^ { 2 } = 4 + c ^ { 2 }\)
2. Hence find the equations of the tangents to \(H\) that pass through the point ( 1,2 ).
3. Find the coordinates of the point of contact each of these tangents makes with \(H\).

1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants.

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.

1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).

7. The ellipse \(E\) has foci at the points \(( \pm 3,0 )\) and has directrices with equations \(x = \pm \frac { 25 } { 3 }\)

1. Find a cartesian equation for the ellipse \(E\). The straight line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
2. Show that the \(x\) coordinates of any points of intersection of \(l\) and \(E\) satisfy the equation $$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$ Given that the line \(l\) is a tangent to \(E\),
3. show that \(c ^ { 2 } = p m ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found. The line \(l\) intersects the \(x\)-axis at the point \(A\) and intersects the \(y\)-axis at the point \(B\).
4. Show that the area of triangle \(O A B\), where \(O\) is the origin, is $$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
5. Find the minimum area of triangle \(O A B\).

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   Q7

   
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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 \(C\).

9 A hyperbola \(H\) has equation $$x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$$

1. Find the equations of the two asymptotes of \(H\), giving each answer in the form \(y = m x\).
2. Draw a sketch of the two asymptotes of \(H\), using roughly equal scales on the two coordinate axes. Using the same axes, sketch the hyperbola \(H\).
3. 1. Show that, if the line \(y = x + c\) intersects \(H\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$x ^ { 2 } - 2 c x - \left( c ^ { 2 } + 2 \right) = 0$$
   2. Hence show that the line \(y = x + c\) intersects \(H\) in two distinct points, whatever the value of \(c\).
   3. Find, in terms of \(c\), the \(y\)-coordinates of these two points.

9 An ellipse is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-5_453_633_365_699} The ellipse intersects the \(x\)-axis at the points \(A\) and \(B\). The equation of the ellipse is $$\frac { ( x - 4 ) ^ { 2 } } { 4 } + y ^ { 2 } = 1$$

1. Find the \(x\)-coordinates of \(A\) and \(B\).
2. The line \(y = m x ( m > 0 )\) is a tangent to the ellipse, with point of contact \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$\left( 1 + 4 m ^ { 2 } \right) x ^ { 2 } - 8 x + 12 = 0$$
   2. Hence find the exact value of \(m\).
   3. Find the coordinates of \(P\).

9 The diagram shows the parabola \(y ^ { 2 } = 4 x\) and the point \(A\) with coordinates \(( 3,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-05_732_657_370_689}

1. Find an equation of the straight line having gradient \(m\) and passing through the point \(A ( 3,4 )\).
2. Show that, if this straight line intersects the parabola, then the \(y\)-coordinates of the points of intersection satisfy the equation $$m y ^ { 2 } - 4 y + ( 16 - 12 m ) = 0$$
3. By considering the discriminant of the equation in part (b), find the equations of the two tangents to the parabola which pass through \(A\).
   (No credit will be given for solutions based on differentiation.)
4. Find the coordinates of the points at which these tangents touch the parabola.

9 A parabola \(P\) has equation \(y ^ { 2 } = x - 2\).

1. 1. Sketch the parabola \(P\).
   2. On your sketch, draw the two tangents to \(P\) which pass through the point \(( - 2,0 )\).
2. 1. Show that, if the line \(y = m ( x + 2 )\) intersects \(P\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$m ^ { 2 } x ^ { 2 } + \left( 4 m ^ { 2 } - 1 \right) x + \left( 4 m ^ { 2 } + 2 \right) = 0$$
   2. Show that, if this equation has equal roots, then $$16 m ^ { 2 } = 1$$
   3. Hence find the coordinates of the points at which the tangents to \(P\) from the point \(( - 2,0 )\) touch the parabola \(P\).

8 The diagram shows the ellipse \(E\) with equation $$\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$$ and the straight line \(L\) with equation $$y = x + 4$$ \includegraphics[max width=\textwidth, alt={}, center]{9f8cd5ed-f5cf-4cf6-8c92-9fd0819238ca-5_675_1120_708_468}

1. Write down the coordinates of the points where the ellipse \(E\) intersects the coordinate axes.
2. The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { c } p \\ 0 \end{array} \right]\), where \(p\) is a constant. Write down the equation of the translated ellipse.
3. Show that, if the translated ellipse intersects the line \(L\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$9 x ^ { 2 } - ( 8 p - 40 ) x + \left( 4 p ^ { 2 } + 60 \right) = 0$$
4. Given that the line \(L\) is a tangent to the translated ellipse, find the coordinates of the two possible points of contact.
   (No credit will be given for solutions based on differentiation.)

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

1. Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes.
   [0pt] [2 marks]
2. Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(- 5 < k < 5\).
   [0pt] [5 marks]
3. The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { l } a \\ b \end{array} \right]\) to form another ellipse whose equation is \(9 x ^ { 2 } + 16 y ^ { 2 } + 18 x - 64 y = c\). Find the values of the constants \(a , b\) and \(c\).
   [0pt] [5 marks]
4. Hence find an equation for each of the two tangents to the ellipse \(9 x ^ { 2 } + 16 y ^ { 2 } + 18 x - 64 y = c\) that are parallel to the line \(y = x\).
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}]{2eaee88a-9e08-4392-8a4c-79fc9861603e-10_1438_1707_1265_153}

6. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants. The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.

1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).

9 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ and its asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-6_798_939_612_555} The constants \(a\) and \(b\) are positive integers.
The point \(A\) on the hyperbola has coordinates ( 2,0 ).
The equations of the asymptotes are \(y = 2 x\) and \(y = - 2 x\).

1. Show that \(a = 2\) and \(b = 4\).
2. The point \(P\) has coordinates ( 1,0 ). A straight line passes through \(P\) and has gradient \(m\). Show that, if this line intersects the hyperbola, the \(x\)-coordinates of the points of intersection satisfy the equation $$\left( m ^ { 2 } - 4 \right) x ^ { 2 } - 2 m ^ { 2 } x + \left( m ^ { 2 } + 16 \right) = 0$$
3. Show that this equation has equal roots if \(3 m ^ { 2 } = 16\).
4. There are two tangents to the hyperbola which pass through \(P\). Find the coordinates of the points at which these tangents touch the hyperbola.
   (No credit will be given for solutions based on differentiation.)