Straight Lines & Coordinate Geometry

8 Find the equation of the line that passes through the point \(( 1,2 )\) and is perpendicular to the line \(3 x + 2 y = 5\).

The line \(L\) has equation \(2x + 3y = 7\) Which one of the following is perpendicular to \(L\)? Tick one box. [1 mark] \(2x - 3y = 7\) \(3x + 2y = -7\) \(2x + 3y = -\frac{1}{7}\) \(3x - 2y = 7\)

9. The line \(L _ { 1 }\) has equation \(4 y + 3 = 2 x\) The point \(A ( p , 4 )\) lies on \(L _ { 1 }\)

1. Find the value of the constant \(p\). The line \(L _ { 2 }\) passes through the point \(C ( 2,4 )\) and is perpendicular to \(L _ { 1 }\)
2. Find an equation for \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(L _ { 1 }\) and the line \(L _ { 2 }\) intersect at the point \(D\).
3. Find the coordinates of the point \(D\).
4. Show that the length of \(C D\) is \(\frac { 3 } { 2 } \sqrt { } 5\) A point \(B\) lies on \(L _ { 1 }\) and the length of \(A B = \sqrt { } ( 80 )\) The point \(E\) lies on \(L _ { 2 }\) such that the length of the line \(C D E = 3\) times the length of \(C D\).
5. Find the area of the quadrilateral \(A C B E\).

5 Find the coordinates of the point of intersection of the lines \(y = 5 x - 2\) and \(x + 3 y = 8\).

The line \(L_1\) passes through the points \(A(-1, 3)\) and \(B(2, 9)\). The line \(L_2\) has equation \(2y + x = 25\) and intersects \(L_1\) at the point \(C\). \(L_2\) also intersects the \(x\)-axis at the point \(D\).

1. Show that the equation of the line \(L_1\) is \(y = 2x + 5\). [3]
2. 1. Find the coordinates of the point \(D\).
   2. Show that \(L_1\) and \(L_2\) are perpendicular.
   3. Determine the coordinates of \(C\). [5]
3. Find the length of \(CD\). [2]
4. Calculate the angle \(ADB\). Give your answer in degrees, correct to one decimal place. [5]

10. The straight line \(l\) has gradient 3 and passes through the point \(A ( - 6,4 )\).

1. Find an equation for \(l\) in the form \(y = m x + c\). The straight line \(m\) has the equation \(x - 7 y + 14 = 0\).
   Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
2. find the coordinates of \(B\) and \(C\),
3. show that \(\angle B A C = 90 ^ { \circ }\),
4. find the area of triangle \(A B C\).

A line has gradient 3 and passes through the point \((1, -5)\). The point \((5, k)\) is on this line. Find the value of \(k\). [2]

A line has gradient 3 and passes through the point \((1, -5)\). The point \((5, k)\) is on this line. Find the value of \(k\). [2]

2. \begin{figure}[h] \end{figure} Figure 1 shows the curve defined by the equation $$y ^ { 2 } + 3 y - 6 \sin y = 4 - x ^ { 2 }$$ The point \(P ( x , y )\) lies on the curve.
The distance from the origin,\(O\) ,to \(P\) is \(D\) .

1. Write down an equation for \(D ^ { 2 }\) in terms of \(y\) only.
2. Hence determine the minimum value of \(D\) giving your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-04_2266_53_312_1977}

1 Find the equation of the line which passes through \(( 1,3 )\) and ( 4,9 ).

1 Find the equation of the line which passes through the points \(( 2,5 )\) and \(( 8 , - 1 )\). Show that this line also passes through the point \(( - 2,9 )\).

10 A triangle has vertices \(A ( 1,4 ) , B ( 7,0 )\) and \(C ( - 4 , - 1 )\).

1. Show that the equation of the line AC is \(\mathrm { y } = \mathrm { x } + 3\). M is the midpoint of AB . The line AC intersects the \(x\)-axis at D .
2. Determine the angle DMA.

1 Find the coordinates of the point at which the perpendicular bisector of the line joining (2, 7) to \(( 10,3 )\) meets the \(x\)-axis.

2 Two points \(A\) and \(B\) have coordinates \(( 1,3 )\) and \(( 9 , - 1 )\) respectively. The perpendicular bisector of \(A B\) intersects the \(y\)-axis at the point \(C\). Find the coordinates of \(C\).

8 A curve has equation \(y = 3 x - \frac { 4 } { x }\) and passes through the points \(A ( 1 , - 1 )\) and \(B ( 4,11 )\). At each of the points \(C\) and \(D\) on the curve, the tangent is parallel to \(A B\). Find the equation of the perpendicular bisector of \(C D\).

Find, in the form \(y = ax + b\), the equation of the line through \((3, 10)\) which is parallel to \(y = 2x + 7\). [3]

1 Find, in the form \(y = a x + b\), the equation of the line through \(( 3,10 )\) which is parallel to \(y = 2 x + 7\).

10. \begin{figure}[h] \end{figure} Figure 1 shows the parallelogram \(A B C D\).
The points \(A\) and \(B\) have coordinates \(( - 1,3 )\) and \(( 3,4 )\) respectively and lie on the straight line \(l _ { 1 }\).

1. Find an equation for \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The points \(C\) and \(D\) lie on the straight line \(l _ { 2 }\) which has the equation \(x - 4 y - 21 = 0\).
2. Show that the distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(k \sqrt { 17 }\), where \(k\) is an integer to be found.
3. Find the area of parallelogram \(A B C D\).

6 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-3_593_878_269_635} The diagram shows a rectangle \(A B C D\). The point \(A\) is \(( 2,14 ) , B\) is \(( - 2,8 )\) and \(C\) lies on the \(x\)-axis. Find

1. the equation of \(B C\),
2. the coordinates of \(C\) and \(D\).

3. \begin{figure}[h] \end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 2.

1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
3. find the value of \(k\),
4. find the coordinates of \(D\).

11 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-4_995_867_260_639} The diagram shows a parallelogram \(A B C D\), in which the equation of \(A B\) is \(y = 3 x\) and the equation of \(A D\) is \(4 y = x + 11\). The diagonals \(A C\) and \(B D\) meet at the point \(E \left( 6 \frac { 1 } { 2 } , 8 \frac { 1 } { 2 } \right)\). Find, by calculation, the coordinates of \(A , B , C\) and \(D\).

A is the point \((1, 5)\) and B is the point \((6, -1)\). M is the midpoint of AB. Determine whether the line with equation \(y = 2x - 5\) passes through M. [3]

The points \(A\) and \(B\) have coordinates \((1, 2)\) and \((5, 8)\) respectively.

1. Find the coordinates of the mid-point of \(AB\). [2]
2. Find, in the form \(y = mx + c\), an equation for the straight line through \(A\) and \(B\). [4]

7 Points \(A\) and \(B\) lie on the curve \(y = x ^ { 2 } - 4 x + 7\). Point \(A\) has coordinates \(( 4,7 )\) and \(B\) is the stationary point of the curve. The equation of a line \(L\) is \(y = m x - 2\), where \(m\) is a constant.

1. In the case where \(L\) passes through the mid-point of \(A B\), find the value of \(m\).
2. Find the set of values of \(m\) for which \(L\) does not meet the curve.

2

1. Find the points of intersection of the line \(2 x + 3 y = 12\) with the axes.
2. Find also the gradient of this line.

2

1. Find the points of intersection of the line \(2 x + 3 y = 12\) with the axes.
2. Find also the gradient of this line.

The line \(L\) is parallel to \(y = -2x + 1\) and passes through the point \((5, 2)\). Find the coordinates of the points of intersection of \(L\) with the axes. [5]

8 The line \(l\) has gradient - 2 and passes through the point \(A ( 3,5 ) . B\) is a point on the line \(l\) such that the distance \(A B\) is \(6 \sqrt { 5 }\). Find the coordinates of each of the possible points \(B\).

1. The line joining the points \((-2, 7)\) and \((-4, p)\) has gradient 4. Find the value of \(p\). [3]
2. The line segment joining the points \((-2, 7)\) and \((6, q)\) has mid-point \((m, 5)\). Find \(m\) and \(q\). [3]
3. The line segment joining the points \((-2, 7)\) and \((d, 3)\) has length \(2\sqrt{13}\). Find the two possible values of \(d\). [4]

8 The line \(l\) has gradient - 2 and passes through the point \(A ( 3,5 ) . B\) is a point on the line \(l\) such that the distance \(A B\) is \(6 \sqrt { 5 }\). Find the coordinates of each of the possible points \(B\).

5 The vertices of a triangle have coordinates ( 1,5 ), ( \(- 3,7\) ) and ( \(- 2 , - 1\) ).
Show that the triangle is right-angled.

4 The coordinates of the points \(\mathrm { A } , \mathrm { B }\) and C are ( \(- 2,2\) ), ( 1,3 ) and ( \(3 , - 3\) ) respectively.

1. Find the gradients of the lines AB and BC .
2. Show that the triangle ABC is a right-angled triangle.
3. Find the area of the triangle ABC .

14

1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
2. Prove that \(h = \sqrt { a b }\), as given in line 24 .

11 Triangle \(A B C\) has vertices at \(A ( - 2 , - 1 ) , B ( 4,6 )\) and \(C ( 6 , - 3 )\).

1. Show that triangle \(A B C\) is isosceles and find the exact area of this triangle.
2. The point \(D\) is the point on \(A B\) such that \(C D\) is perpendicular to \(A B\). Calculate the \(x\)-coordinate of \(D\).

5 A line has gradient - 4 and passes through the point (2,6). Find the coordinates of its points of intersection with the axes. \begin{figure}[h] \end{figure} Fig. 11 shows the line joining the points \(\mathrm { A } ( 0,3 )\) and \(\mathrm { B } ( 6,1 )\).

1. Find the equation of the line perpendicular to AB that passes through the origin, O .
2. Find the coordinates of the point where this perpendicular meets AB .
3. Show that the perpendicular distance of AB from the origin is \(\frac { 9 \sqrt { 10 } } { 10 }\).
4. Find the length of AB , expressing your answer in the form \(a \sqrt { 10 }\).
5. Find the area of triangle OAB .

11 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-4_563_965_813_591} In the diagram, the points \(A\) and \(C\) lie on the \(x\) - and \(y\)-axes respectively and the equation of \(A C\) is \(2 y + x = 16\). The point \(B\) has coordinates ( 2,2 ). The perpendicular from \(B\) to \(A C\) meets \(A C\) at the point \(X\).

1. Find the coordinates of \(X\). The point \(D\) is such that the quadrilateral \(A B C D\) has \(A C\) as a line of symmetry.
2. Find the coordinates of \(D\).
3. Find, correct to 1 decimal place, the perimeter of \(A B C D\).

Determine whether the line with equation \(2x + 3y + 4 = 0\) is parallel to the line through the points with coordinates \((9, 4)\) and \((3, 8)\). [4 marks]

5 The curve \(y ^ { 2 } = 12 x\) intersects the line \(3 y = 4 x + 6\) at two points. Find the distance between the two points.

13 Show that the larger regular hexagon in Fig. C1 has perimeter \(4 \sqrt { 3 }\).

Find the gradient of the line with equation \(2x + 5y = 7\) Circle your answer. [1 mark] \(\frac{2}{5}\) \quad \(\frac{5}{2}\) \quad \(-\frac{2}{5}\) \quad \(-\frac{5}{2}\)

A line \(L\) is parallel to the line \(x + 2y = 6\) and passes through the point \((10, 1)\). Find the area of the region bounded by the line \(L\) and the axes. [5]

8 The equation of a curve is \(y = 5 - \frac { 8 } { x }\).

1. Show that the equation of the normal to the curve at the point \(P ( 2,1 )\) is \(2 y + x = 4\). This normal meets the curve again at the point \(Q\).
2. Find the coordinates of \(Q\).
3. Find the length of \(P Q\).

1 Find the value of the constant \(c\) for which the line \(y = 2 x + c\) is a tangent to the curve \(y ^ { 2 } = 4 x\).

The line with gradient \(-2\) passing through the point \(P(3t, 2t)\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Find the area of triangle \(AOB\) in terms of \(t\). [3]

The line through \(P\) perpendicular to \(AB\) intersects the \(x\)-axis at \(C\).

1. Show that the mid-point of \(PC\) lies on the line \(y = x\). [4]

11 \includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-4_885_967_255_589} The diagram shows the curve \(y = ( 6 x + 2 ) ^ { \frac { 1 } { 3 } }\) and the point \(A ( 1,2 )\) which lies on the curve. The tangent to the curve at \(A\) cuts the \(y\)-axis at \(B\) and the normal to the curve at \(A\) cuts the \(x\)-axis at \(C\).

1. Find the equation of the tangent \(A B\) and the equation of the normal \(A C\).
2. Find the distance \(B C\).
3. Find the coordinates of the point of intersection, \(E\), of \(O A\) and \(B C\), and determine whether \(E\) is the mid-point of \(O A\).

The points \(A\) and \(B\) have coordinates \(( - 1,10 )\) and \(( 5,1 )\) respectively. The straight line \(L\) has equation \(2 x - 3 y + 6 = 0\). a) The line \(L\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).
b) Determine the ratio in which the line \(L\) divides the line \(A B\).
c) The line \(L\) crosses the \(x\)-axis at the point \(D\). Find the coordinates of \(D\).
d) i) Show that \(L\) is perpendicular to \(A B\).
ii) Calculate the area of the triangle \(A C D\).

7 The point \(R\) is the reflection of the point \(( - 1,3 )\) in the line \(3 y + 2 x = 33\). Find by calculation the coordinates of \(R\).

\includegraphics{figure_9} The diagram shows the parallelogram \(ABCD\). The points \(A\) and \(B\) have coordinates \((-1, 3)\) and \((3, 4)\) respectively and lie on the straight line \(l_1\).

1. Find an equation for \(l_1\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The points \(C\) and \(D\) lie on the straight line \(l_2\) which has the equation \(x - 4y - 21 = 0\).

1. Show that the distance between \(l_1\) and \(l_2\) is \(k\sqrt{17}\), where \(k\) is an integer to be found. [7]
2. Find the area of parallelogram \(ABCD\). [2]

4. A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
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