📖 Hover (or tap) any dotted word for its definition. Write your answers in the squared boxes with a pencil, finger or mouse — they save on your device.
(a)The circle c has equation x2 + y2 + 4x − 10y − 52 = 0. Write down the centre and radius of the circle c.
(a) — centre & radius
(b)Three standard unbiased 6-sided dice are labelled h, k, and r. Gino rolls the three dice and uses the numbers shown to form the equation of a circle in the form (x − h)2 + (y − k)2 = r2, where h, k, and r are the numbers shown on each of the dice. Each different set of values for h, k, and r gives a different circle.
(i)Write down an example of one set of numbers h, k, and r that Gino could throw, and write down the equation of the circle formed using these numbers.
(b)(i)
(ii)In total, how many different circles can Gino make in this way?
(b)(ii)
(iii)How many of these circles have both the x-axis and the y-axis as a tangent?
(b)(iii)
(iv)How many of these circles lie entirely inside the first quadrant? (That is, how many neither touch nor cross the x-axis nor the y-axis?)
(b)(iv)
(a)A circle s has the equation (x − 4)2 + (y + 2)2 = 45.
(i)Write down the centre and radius of the circle s.
(a)(i)
(ii)Find the equation of the tangent to s at the point (−2, −5). Write your answer in the form y = mx + c, where m, c ∈ ℤ.
(a)(ii)
(b)The circle t has the following equation, where k ∈ ℝ is a constant: x2 + y2 + 28x − 46y + k = 0. The horizontal line y = k is a tangent to the circle t. Find the two possible values of k.
(b)
A submarine is within √58 km of a point A and is within √178 km of a point B. The distance from A to B is 20 km. A, B, and the submarine are all at the same depth. This is represented on the co-ordinate diagram below. The points A(0, 0) and B(20, 0) are shown. Part of the circles k and s are shown, where k has centre A(0, 0) and radius √58, and s has centre B(20, 0) and radius √178. The circles intersect at the points C and D. The submarine is in the shaded region, R, the region that is inside both circles k and s.
(i)Write down the equation of the circle s.
(c)(i)
The point C is (7, 3) and the point D is (7, −3).
(ii)By using your answer to part (c)(i), or otherwise, verify that (7, 3) lies on the circle s.
(c)(ii)
(iii)By using C(7, 3) and D(7, −3), find the area of the triangle DBC, in km2.
(c)(iii)
(iv)Find the size of the acute angle ∠CBD, correct to the nearest degree.
(c)(iv)
(v)The area of the sector ADC of the circle k is 23·4837 km2, correct to 4 decimal places. The area of the triangle ADC is 21 km2. Using this, work out the area of the shaded region, R, that is inside both circles. Give your answer in km2, correct to 2 decimal places.
(c)(v)
(a)The circle s has equation x2 + y2 + 4x − 6y + 5 = 0.
(i)Write down the centre and radius of the circle s.
(a)(i)
(ii)The circle c has equation (x − 2)2 + (y + 1)2 = 72. Show that the circles s and c touch internally.
(a)(ii)
(b)Another circle has its centre on the vertical line through the point (9, 0). The points (7, 10) and (12, 8) are on this circle. Find the equation of this circle. Note that your answer may contain non-integer values.
(b)
Ameena, Petro, and Fiadh are taking part in an adventure race. The co-ordinate diagram below shows part of the course for this race. Each unit on the diagram represents 100 metres.
(a)The arc in the diagram represents part of a stream. The arc is part of a circle s with centre C(1, 17) and radius 12.
(i)Write down the equation of the circle s.
(a)(i)
(ii)Ameena is at the point (a, 8) on the stream (circle s), where a ∈ ℝ, a > 0. Work out the value of a. Give your answer in surd form.
(a)(ii)
(iii)Petro is at the point P(10, 6). Work out the shortest distance from the point P to the stream (circle s). Give your answer correct to the nearest metre. Remember that each unit on the diagram represents 100 metres.
(a)(iii)
(b)There is a straight path, l, that is not shown on the diagram. l is parallel to the y-axis, and is a tangent to the stream s in the first quadrant. Write down the equation of this path l. Remember that the radius of s is 12.
(b)
(a)The circle c has equation (x − h)2 + (y + 3)2 = 12, where h ∈ ℝ.
(i)Write down the centre and radius of the circle c. Give your answer in terms of h, where appropriate.
(a)(i)
(ii)The perpendicular distance from the line x − 4y + 7 = 0 to the centre of the circle c is 5 units. Work out the two possible values of h. Give each answer in surd form.
(a)(ii)
(b)The circle s passes through the points (8, 1), (a, 3), and (a, −5), as shown in the diagram (not to scale), where 0 < a < 5, a ∈ ℝ. The radius of the circle s is √20. Find the equation of the circle s.
(b)
(c)Ava looks at a tiling of the inside of the unit circle c : x2 + y2 = 1. The diagram (not to scale) shows the circle c and another circle, s. The points P and Q are on both circles. The part of s that lies inside c is an edge of a number of tiles. Ava wants to find the equation of the circle s.
(i)|∠QOP| = 45°, where O is the point (0, 0). Show that the point Q has co-ordinates (−1√2, 1√2).
(c)(i)
(ii)The point P lies on the x-axis. The centre of the circle s lies on the tangent to c at the point P and on the tangent to c at the point Q. Find the centre and the radius of the circle s. Give your answers in surd form.
(c)(ii)
(a)The circle c has equation x2 + y2 − 2x + 8y + k = 0, where k ∈ ℝ. The radius of c is 5√3. Find the value of k.
(a)
(b)The circle (x − 5)2 + (y + 2)2 = 20 has a tangent at the point (9, −4). Find the slope of this tangent.
(b)
(c)Two circles each have both the x-axis and the y-axis as tangents, and each contains the point (1, −8), as shown (not to scale). Find the equation of each of these circles.
(c)
(a)The circle k has centre C(1, −2) and chord [AB] where |AB| = 4√3. The point D(3, 2) is the midpoint of the chord [AB] (shown). Find the radius of k. Give your answer in the form a√b, where a, b ∈ ℕ.
(a)
(b)(i) Show that the circles c : x2 + y2 + 4x − 2y − 95 = 0 and s : (x − 7)2 + (y − 13)2 = 25 touch externally.
(b)(i)
(ii)There are an infinite number of circles which touch circle c externally at the same point that s touches c. Find the coordinates of the centre of one of these circles, apart from circle s.
(b)(ii)
(a)The circle c has equation x2 + y2 − 4x + 2y − 4 = 0. The point A is the centre of the circle. The line l is a tangent to c at the point T (shown). The point B(5, 8) is on l. Find |BT|.
(a)
(b)Two circles, c1 and c2, have their centres on the x-axis. Each circle has a radius of 5 units. The point (1, 4) lies on each circle. Find the equation of c1 and the equation of c2.
(b)
(a)The point (−2, k) is on the circle (x − 2)2 + (y − 3)2 = 65. Find the two possible values of k, where k ∈ ℤ.
(a)
(b)The circle s is in the first quadrant. It touches both the x-axis and the y-axis. The line t: 3x − 4y + 6 = 0 is a tangent to s as shown. Find the equation of s.
(b)
The lines are m: 2x + 3y + 1 = 0 and n: 2x + 3y − 51 = 0.
(c)Two touching circles, s and t, are shown. m is a tangent to s at A and n is a tangent to t at B. The ratio of the radius of s to the radius of t is 1 : 3. Find the equation of s.
(c)
A(0, 0), B(6·5, 0) and C(10, 7) are three points on a circle.
(a)Find the equation of the circle.
(a)
(b)Find |∠BCA|. Give your answer in degrees, correct to 2 decimal places.
(b)
A point X(−1, 6) lies on the line XC of slope 17.
(b)C is the centre of a circle s of radius 5 cm. The line l: 3x + 4y − 21 = 0 is a tangent to s and passes through X, as shown. Find the equation of one such circle s.
(b)
Two circles s and c touch internally at B, as shown.
(a)The equation of circle s is (x − 1)2 + (y + 6)2 = 360. Write down the co-ordinates of the centre of s, and the radius of s in the form a√10, where a ∈ ℕ.
(a)
(b)(i)The point K is the centre of circle c. The radius of c is one-third the radius of s. The co-ordinates of B are (7, 12). Find the co-ordinates of K.
(b)(i)
(ii)Find the equation of c.
(b)(ii)
(c)Find the equation of the common tangent at B. Give your answer in the form ax + by + c = 0, where a, b, c ∈ ℤ.
(c)
The diagram shows a circular clock face. The square part of the clock face is glass so that the mechanism is visible. Two circular cogs, h and k, which touch externally, are shown. The point C is the centre of the clock face, D is the centre of the larger cog h, and E is the centre of the smaller cog k.
(i)In suitable co-ordinates, the equation of the circle h is x2 + y2 + 4x + 6y − 19 = 0. Find the radius of h, and the co-ordinates of its centre, D.
(i)
(ii)The point E has co-ordinates (3, 2). Find the radius of the circle k.
(ii)
(iii)Show that the distance from C(−2, 2) to the line DE is half the length of [DE].
(iii)
(iv)The translation which maps the midpoint of DE to the point C maps the circle k to the circle j. Find the equation of the circle j.
(iv)
(v)The glass square is of side length l. Find the smallest whole number l such that the two cogs, h and k, are fully visible through the glass.
(v)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.