1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

10

1. The coordinates of two points \(A\) and \(B\) are \(( - 7,3 )\) and \(( 5,11 )\) respectively.
   Show that the equation of the perpendicular bisector of \(A B\) is \(3 x + 2 y = 11\).
2. A circle passes through \(A\) and \(B\) and its centre lies on the line \(12 x - 5 y = 70\). Find an equation of the circle.

6 Points \(A\) and \(B\) have coordinates \(( 8,3 )\) and \(( p , q )\) respectively. The equation of the perpendicular bisector of \(A B\) is \(y = - 2 x + 4\). Find the values of \(p\) and \(q\).

7 The point \(A\) has coordinates \(( 1,5 )\) and the line \(l\) has gradient \(- \frac { 2 } { 3 }\) and passes through \(A\). A circle has centre \(( 5,11 )\) and radius \(\sqrt { 52 }\).

1. Show that \(l\) is the tangent to the circle at \(A\).
2. Find the equation of the other circle of radius \(\sqrt { 52 }\) for which \(l\) is also the tangent at \(A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-10_887_1003_258_571} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 20\) and with centre \(C\). The point \(B\) has coordinates \(( 0,2 )\) and the line segment \(B C\) intersects the circle at \(P\).

1. Find the equation of \(B C\).
2. Hence find the coordinates of \(P\), giving your answer in exact form.

11 The point \(P\) lies on the line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. A curve has equation \(y = - \frac { m } { x }\). There is a single point \(P\) on the curve such that the straight line is a tangent to the curve at \(P\).

1. Find the coordinates of \(P\), giving the \(y\)-coordinate in terms of \(m\).
   The normal to the curve at \(P\) intersects the curve again at the point \(Q\).
2. Find the coordinates of \(Q\) in terms of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

12 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-18_1006_938_269_591} The diagram shows a circle \(P\) with centre \(( 0,2 )\) and radius 10 and the tangent to the circle at the point \(A\) with coordinates \(( 6,10 )\). It also shows a second circle \(Q\) with centre at the point where this tangent meets the \(y\)-axis and with radius \(\frac { 5 } { 2 } \sqrt { 5 }\).

1. Write down the equation of circle \(P\).
2. Find the equation of the tangent to the circle \(P\) at \(A\).
3. Find the equation of circle \(Q\) and hence verify that the \(y\)-coordinates of both of the points of intersection of the two circles are 11.
4. Find the coordinates of the points of intersection of the tangent and circle \(Q\), giving the answers in surd form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A circle has equation \(( x - 1 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 40\). A line with equation \(y = x - 9\) intersects the circle at points \(A\) and \(B\).

1. Find the coordinates of the two points of intersection.
2. Find an equation of the circle with diameter \(A B\).

5 Points \(A ( 7,12 )\) and \(B\) lie on a circle with centre \(( - 2,5 )\). The line \(A B\) has equation \(y = - 2 x + 26\).
Find the coordinates of \(B\).

11 \includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-16_599_780_274_671} The diagram shows the curve with equation \(x = y ^ { 2 } + 1\). The points \(A ( 5,2 )\) and \(B ( 2 , - 1 )\) lie on the curve.

1. Find an equation of the line \(A B\).
2. Find the volume of revolution when the region between the curve and the line \(A B\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-12_583_661_260_740} The diagram shows a circle with centre \(A\) passing through the point \(B\). A second circle has centre \(B\) and passes through \(A\). The tangent at \(B\) to the first circle intersects the second circle at \(C\) and \(D\). The coordinates of \(A\) are ( \(- 1,4\) ) and the coordinates of \(B\) are ( 3,2 ).

1. Find the equation of the tangent CBD.
2. Find an equation of the circle with centre \(B\).
3. Find, by calculation, the \(x\)-coordinates of \(C\) and \(D\). \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-14_746_652_262_744} The diagram shows a sector \(C A B\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D , E\) and \(F\), where \(C O D\) is a straight line and angle \(A C D\) is \(\theta\) radians.

7 A circle with centre \(( 5,2 )\) passes through the point \(( 7,5 )\).

1. Find an equation of the circle.
   The line \(y = 5 x - 10\) intersects the circle at \(A\) and \(B\).
2. Find the exact length of the chord \(A B\).

12 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-18_750_981_258_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y - 27 = 0\) and the tangent to the circle at the point \(P ( 5,4 )\).

1. The tangent to the circle at \(P\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). Find the area of triangle \(O A B\), where \(O\) is the origin.
2. Points \(Q\) and \(R\) also lie on the circle, such that \(P Q R\) is an equilateral triangle. Find the exact area of triangle \(P Q R\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 The line \(y = 2 x + 5\) intersects the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\) at \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\) in surd form and hence find the exact length of the chord \(A B\).
   A straight line through the point \(( 10,0 )\) with gradient \(m\) is a tangent to the circle.
2. Find the two possible values of \(m\).

1 Points \(A\) and \(B\) have coordinates \(( 5,2 )\) and \(( 10 , - 1 )\) respectively.

1. Find the equation of the perpendicular bisector of \(A B\).
2. Find the equation of the circle with centre \(A\) which passes through \(B\).

11 The coordinates of points \(A , B\) and \(C\) are \(A ( 5 , - 2 ) , B ( 10,3 )\) and \(C ( 2 p , p )\), where \(p\) is a constant.

1. Given that \(A C\) and \(B C\) are equal in length, find the value of the fraction \(p\).
2. It is now given instead that \(A C\) is perpendicular to \(B C\) and that \(p\) is an integer.
   1. Find the value of \(p\).
   2. Find the equation of the circle which passes through \(A , B\) and \(C\), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are constants.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 The circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 20 = 0\) has centre \(C\) and passes through points \(A\) and \(B\).

1. State the coordinates of \(C\).
   It is given that the midpoint, \(D\), of \(A B\) has coordinates \(\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)\).
2. Find the equation of \(A B\), giving your answer in the form \(y = m x + c\).
3. Find, by calculation, the \(x\)-coordinates of \(A\) and \(B\).

7 The line \(L _ { 1 }\) has equation \(2 x + y = 8\). The line \(L _ { 2 }\) passes through the point \(A ( 7,4 )\) and is perpendicular to \(L _ { 1 }\).

1. Find the equation of \(L _ { 2 }\).
2. Given that the lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\), find the length of \(A B\).

6 The curve \(y = 9 - \frac { 6 } { x }\) and the line \(y + x = 8\) intersect at two points. Find

1. the coordinates of the two points,
2. the equation of the perpendicular bisector of the line joining the two points.

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.

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 \(D\).

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 \(A B C D\).

8 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_599_716_1071_717} The diagram shows points \(A , B\) and \(C\) lying on the line \(2 y = x + 4\). The point \(A\) lies on the \(y\)-axis and \(A B = B C\). The line from \(D ( 10 , - 3 )\) to \(B\) is perpendicular to \(A C\). Calculate the coordinates of \(B\) and \(C\).

8 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_625_547_1489_797} The diagram shows a triangle \(A B C\) in which \(A\) is \(( 3 , - 2 )\) and \(B\) is \(( 15,22 )\). The gradients of \(A B , A C\) and \(B C\) are \(2 m , - 2 m\) and \(m\) respectively, where \(m\) is a positive constant.

1. Find the gradient of \(A B\) and deduce the value of \(m\).
2. Find the coordinates of \(C\). The perpendicular bisector of \(A B\) meets \(B C\) at \(D\).
3. Find the coordinates of \(D\).

8 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_796_695_1539_726} The diagram shows a rhombus \(A B C D\) in which the point \(A\) is ( \(- 1,2\) ), the point \(C\) is ( 5,4 ) and the point \(B\) lies on the \(y\)-axis. Find

1. the equation of the perpendicular bisector of \(A C\),
2. the coordinates of \(B\) and \(D\),
3. the area of the rhombus.

10

1. Express \(2 x ^ { 2 } - 4 x + 1\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2 x ^ { 2 } - 4 x + 1\). The line \(x - y + 4 = 0\) intersects the curve \(y = 2 x ^ { 2 } - 4 x + 1\) at points \(P\) and \(Q\). It is given that the coordinates of \(P\) are \(( 3,7 )\).
2. Find the coordinates of \(Q\).
3. Find the equation of the line joining \(Q\) to the mid-point of \(A P\).