circle - · pdf filethe point p(2,3) lies on the circle (x +1)2 +(y 1)2 = 13. find the...

Higher Mathematics

PSfrag replacementsOxy

Circle

Paper 1 Section A

Each correct answer in this section is worth two marks.

1. A circle has equation (x − 3)2 + (y + 4)2 = 20.

Find the gradient of the tangent to the circle at the point (1, 0) .

A. −2

B. − 12

C. 12

D. 2

Key Outcome Grade Facility Disc. Calculator Content SourceC 2.4 C 0.43 0.77 NC G9, G2, G5 HSN 097


hsn.uk.net


Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes

Higher Mathematics


2. The point (2,−3) lies on the circle with equation x2 + y2 + 6x − 2y + c = 0.

What is the value of c?

A. −31

B. −13

C. −1

D. 9

Key Outcome Grade Facility Disc. Calculator Content SourceA 2.4 C 0.62 0.57 CN G10, A6 HSN 065


3. A circle has centre (2, 4) and passes through (−1, 1) .

What is the equation of the circle?

A. (x − 2)2 + (y − 4)2 =√

18

B. (x − 2)2 + (y − 4)2 = 18

C. (x + 2)2 + (y + 4)2 = 18

D. (x + 2)2 + (y + 4)2 = 26

Key Outcome Grade Facility Disc. Calculator Content SourceB 2.4 C 0.51 0.17 NC G10, G9 HSN 063


hsn.uk.net



Higher Mathematics


4. The point P(−2, 4) lies on the circle with equation x2 + y2 − 2x + 2y − 32 = 0.

What is the gradient of the tangent to the circle at P?

A. 13

B. 35

C. 1

D. 3

Key Outcome Grade Facility Disc. Calculator Content SourceB 2.4 C 0.7 0.06 NC G11 HSN 056


5. A circle has equation (x + 1)2 + (y − 2)2 = 29.

What is the gradient of the tangent to the circle at the point (1,−3)?

A. 25

B. 0

C. − 52

D. − 12

Key Outcome Grade Facility Disc. Calculator Content SourceA 2.4 C 0.49 0.48 NC G11 HSN 014


hsn.uk.net



Higher Mathematics


6. A circle has equation x2 + y2 − 2x − 4y + 1 = 0.

Here are two statements about the circle:

I. The circle has centre (−2,−4) .

II. The circle has radius 1.

Which of the following is true?

A. neither statement is correct

B. only statement I is correct

C. only statement II is correct

D. both statements are correct

Key Outcome Grade Facility Disc. Calculator Content SourceA 2.4 C 0.77 0.57 CN G9 HSN 072


hsn.uk.net



Higher Mathematics


7. A circle has equation x2 + y2 − 4x + 6y + 4 = 0.

Here are two statements about the circle:

I. The circle has centre (−2, 3) .

II. The circle has radius 3 units.

Which of the following is true?

A. neither statement is correct

B. only statement I is correct

C. only statement II is correct

D. both statements are correct

Key Outcome Grade Facility Disc. Calculator Content SourceC 2.4 C 0.47 0.64 NC G9 HSN 076


hsn.uk.net



Higher Mathematics


8. A circle has equation x2 + y2 − ax + 2by + c = 0. The centre of the circle is (−1, 4) .

What are the values of a and b?

a bA. 2 −4

B. −1 −2

C. −2 −4

D. 2 4

Key Outcome Grade Facility Disc. Calculator Content SourceC 2.4 C 0.33 0.23 NC G9 HSN 19


hsn.uk.net



Higher Mathematics


9. A circle has centre (2,−1) , and has the y-axis as a tangent.

What is the equation of the circle?

A. (x + 2)2 + (y − 1)2 = 4

B. (x − 2)2 + (y + 1)2 = 4

C. (x + 2)2 + (y − 1)2 = 1

D. (x − 2)2 + (y + 1)2 = 1



hsn.uk.net



Higher Mathematics


10. What is the largest range of values of k for which the equationx2 + y2 − 6x + 4y + k = 0 represents a circle?

A. k < 52

B. k < 13

C. k > −13

D. All real k



[END OF PAPER 1 SECTION A]

Paper 1 Section B

11.[SQA] The point P(2, 3) lies on the circle (x + 1)2 + (y − 1)2 = 13. Find the equation ofthe tangent at P. 4

Part Marks Level Calc. Content Answer U2 OC44 C CN G11 2y + 3x = 12 2002 P1 Q1

•1 ic: interpret centre from equ. ofcircle

•2 ss: know to find gradient of radius•3 ss: know to find perp. gradient•4 ic: state equation of tangent

•1 C = (−1, 1)•2 mrad = 2

3•3 mtgt = − 3

2•4 y − 3 = − 3

2 (x − 2)

hsn.uk.net



Higher Mathematics


12.[SQA] Circle P has equation x2 + y2 − 8x − 10y + 9 = 0. Circle Q has centre (−2,−1) andradius 2

√2.

(a) (i) Show that the radius of circle P is 4√

2.(ii) Hence show that circles P and Q touch. 4

(b) Find the equation of the tangent to the circle Q at the point (−4, 1) . 3

(c) The tangent in (b) intersects circle P in two points. Find the x -coordinates ofthe points of intersection, expressing you answers in the form a ± b

√3. 3

Part Marks Level Calc. Content Answer U2 OC4(a) 2 C CN G9 proof 2001 P1 Q11(a) 2 A/B CN G14(b) 3 C CN G11 y = x + 5(c) 3 C CN G12 x = 2 ± 2

√3

•1 ic: interpret centre of circle (P)•2 ss: find radius of circle (P)•3 ss: find sum of radii•4 pd: compare with distance between

centres

•5 ss: find gradient of radius•6 ss: use m1m2 = −1•7 ic: state equation of tangent

•8 ss: substitute linear into circle•9 pd: express in standard form•10 pd: solve (quadratic) equation

•1 CP = (4, 5)•2 rP =

√16 + 25 − 9 =

√32 = 4

√2

•3 rP + rQ = 4√

2 + 2√

2 = 6√

2•4 CPCQ =

√62 + 62 = 6

√2 and “so

touch”

•5 mr = −1•6 mtgt = +1•7 y − 1 = 1(x + 4)

•8 x2 + (x + 5)2 − 8x − 10(x + 5) + 9 = 0•9 2x2 − 8x − 16 = 0•10 x = 2 ± 2

√3

hsn.uk.net



Higher Mathematics


13. The diagram below shows thegraph of the cubic with equationy = x3 − 3x2 + 5x + 4 and a circle withcentre C.At the point P the line l is a tangent toboth the curve and the circle.(a) The tangent line l has gradient 2.

Find the coordinates of P. 5

(b) The circle has equationx2 + y2 − 14x − 8y + c = 0. 2Determine the value of c .

(c) The line PQ is a diameter of thecircle. Determine the coordinatesof Q. 2

PSfrag replacements

O x

y

P

Q

C

l

y = x3 − 3x2 + 5x + 4

Part Marks Level Calc. Content Answer U2 OC4(a) 5 C CN C4 P(1, 7) AT059(b) 2 C CN G10 c = 20(c) 2 C CN G9, G6 Q(13, 1)

•1 ss: know to differentiate•2 pd: differentiate•3 ss: equate•4 pd: process solution•5 ic: complete coordinates

•6 ss: substitute•7 pd: process

•8 ic: interpret coordinates•9 ss: use midpoint relationship

•1 dy/dx = · · ·

•2 · · · = 3x2 − 6x + 5•3 3x2 − 6x + 5 = 2•4 x = 1•5 P(1, 7)

•6 1 + 72 − 14 − 8 × 7 + c = 0•7 c = 20

•8 C(7, 4)•9 Q(13, 1)

hsn.uk.net



Higher Mathematics


14. 2+y2 − ax + 2by + c = 0, where a , b and c are constants.

(a) Given that the centre of the circle is (−1, 4) , determine the values of a and b . 2

(b) Given that the point (5, 4) lies on the circle, determine the value of c . 2

Part Marks Level Calc. Content Answer U2 OC4(a) 2 C CN G10 a = −2, b = −4 OB 08-008(b) 2 C CN G10 c = −19

•1 ic: interpret a•2 ic: interpret b

•3 ss: substitute point into circle•4 pd: process for c

•1 a = −2•2 b = −4

•3 52 + 42 + 2 × 5 − 8 × 4 + c = 0•4 c = −19

15. A circle has P(0, 1) and Q(−10, 9) as endpoints of its diameter.

Find the equation of this circle. 3

Part Marks Level Calc. Content Answer U2 OC43 C CN G10, G9 (x + 5)2 + (y − 5)2 = 41 OB 08-001

•1 ss: find midpoint•2 ss: find radius•3 ic: substitute in general form, and

complete

•1 centre is (−5, 5)•2 radius =

√41

•3 (x + 5)2 + (y − 5)2 = 41

16. Find the equation of the circle centred at (1,−3) and passing through (7, 5) .

Determine whether or not the point (8, 4) lies within this circle. 3

Part Marks Level Calc. Content Answer U2 OC43 C CN G10, G9, G1 (x − 1)2 + (y + 3)2 = 100,

withinOB 08-003

•1 ss: find radius•2 ic: state eqn. of circle•3 ic: interpret point

•1 r = 10•2 (x − 1)2 + (y + 3)2 = 100•3 (8 − 1)2 + (4 − 3)2

< 100, so within

hsn.uk.net



Higher Mathematics


17. A circle has equation x2 + y2 − 8x + 6y + 15 = 0.

Find the equation of the tangent to the circle at the point A(1,−2) . 4

Part Marks Level Calc. Content Answer U2 OC44 C CN G11 3x − y − 5 = 0 Ex 2-4-3

•1 ic: interpret circle equation•2 pd: find gradient of radius•3 ss: find perp. gradient•4 ic: state equation of tangent

•1 centre C(4,−3)•2 mAC = − 1

3•3 mtgt = 3•4 y + 2 = 3(x − 1)

18. (a) Find the equation of the line which passes through (−1,−4) and isperpendicular to the line with equation x + 3y = 2. 3

(b) Show that the line does not intersect the circle with equation(x − 2)2 + (y + 3)2 = 6. 3

Part Marks Level Calc. Content Answer U2 OC4(a) 3 C CN G3, G2, G5 3x − y − 1 = 0 AT022(b) 3 C CN G12, A17 proof

•1 ic: express in standard form•2 ss: use m × m⊥ = −1•3 ic: state equation of line

•4 ss: start to solve simultaneously•5 pd: express in standard form•6 ic: use the discriminant

•1 y = − 13 x + 2

3•2 m⊥ = 3•3 y + 4 = 3(x + 1)

•4 (x − 2)2 + (3x − 1 + 3)2 = 6•5 5x2 + 4x + 1 = 0•6

∆ = −4 < 0, so no real roots

19. For which values of k is the line with equation y = 34 x + k a tangent to the circle

with equation x2 + y2 = 16? 5

Part Marks Level Calc. Content Answer U2 OC45 A CN G13 k = ±5 Ex 2-4-6

•1 ss: substitute•2 pd: arrange in standard form•3 ss: know to use “discriminant = 0”•4 ic: identify “a”, “b” and “c”•5 pd: process

•1 x2 + ( 34 x + k)2 = 16

•2 2516 x2 + 3

2 kx + k2 − 16 = 0•3 for tangency, b2 − 4ac = 0•4 a = 25

16 , b = 32 k, c = k2 − 16

•5 ( 32 k)2 − 4 × 25

16 (k2 − 16) = 0and k = ±5

hsn.uk.net



Higher Mathematics


20. The points A and B have coordinates (6, 8) and (18,−8) respectively.

(a) Find the equation of l , the perpendicular bisector of AB. 4

(b) Show that l is a tangent to the circle with equation x2 + y2 − 2x + 4y − 20 = 0and find the coordinates of the point of contact. 6

Part Marks Level Calc. Content Answer U2 OC4(a) 4 C CN G7 3x − 4y − 36 = 0 AT052(b) 6 C CN G13, G12 (4,−6)

•1 ss: determine midpoint•2 pd: determine gradient•3 ss: use m × m⊥ = −1•4 ic: state equation of line

•5 ss: know to use substitution•6 pd: process•7 ic: arrange in standard form•8 ss: strategy for showing tangency•9 ic: complete•10 ic: state coordinates

•1 midptAB = (12, 0)•2 mAB = − 4

3•3 m⊥ = 3

4•4 y = 3

4 (x − 12)

•5 x2 +( 34 (x− 12))2 − 2x + 4( 3

4 (x− 12))− 20 = 0•6 25

16 x2 − 252 x + 25 = 0

•7 x2 − 8x + 16 = 0•8 (x − 4)2 = 0.•9 x = 4•10 (4,−6)

21. For which values of k does the equation x2 + y2 + 10x − 14y + k = 0 represent acircle? 3

Part Marks Level Calc. Content Answer U2 OC43 A CN G9, A6 k < 74 Ex 2-4-4

•1 ss: know to examine radius•2 ic: form inequality•3 pd: process

•1 radius:√

52 + (−7)2 − k•2 25 + 49 − k > 0•3 k < 74

hsn.uk.net



Higher Mathematics


22. Find the range of values of k for which the equation x2 + y2 − kx− (k + 1)y + 54 = 0

represents a circle. 6

Part Marks Level Calc. Content Answer U2 OC46 A CN G9, A16 k < −2, k > 1 AT026

•1 ss: know to use radius•2 ic: interpret radius•3 pd: obtain standard form•4 ss: factorise quadratic•5 ss: method to solve inequality•6 pd: complete

•1 g2 + f 2 − c > 0•2 (− k

2 )2 − (− k+12 )2 − 5

4 > 0•3 k2 + k − 2 > 0•4 (k + 2)(k − 1) > 0•5 e.g. sketch of LHS•6 k < −2, k > 1

23. The circle with centre (8, a) passes through the points (3, 3) and (5, 1) .

Find the value of a and hence find the equation of the circle. 5

Part Marks Level Calc. Content Answer U2 OC45 B CN G9, G10 a = 6,

(x − 8)2 + (y − 6)2 = 34OB 08-004

•1 ss: two expressions for the radius•2 pd: equate and start process for a•3 pd: complete for a•4 pd: calculate radius•5 ic: state eqn. of circle

•1 r2 = (3 − 8)2 + (3 − a)2

and r2 = (5 − 8)2 + (1 − a)2

•2 (3− 8)2 +(3− a)2 = (5− 8)2 +(1− a)2

and expand•3 a = 6•4 radius =

√34

•5 (x − 8)2 + (y − 6)2 = 34

24. (a) Find the radius and equation of the circle centred at the origin and passingthrough the point (3,−4) . 2

(b) Determine whether the point (8,−5) lies inside, outside or on this circle. 2

Part Marks Level Calc. Content Answer U2 OC4(a) 2 C CN G9, G10 x2 + y2 = 25 Ex 2-4-1(b) 2 C CN A9 outside

•1 pd: compute radius•2 ic: state equation

•3 ss: test point•4 ic: conclusion

•1 r =√

32 + (−4)2 = 5•2 x2 + y2 = 25

•3 d2 = 82 + (−5)2 = 89•4 d2

> r2, so the point lies outside thecircle

hsn.uk.net



Higher Mathematics


25. Find the range of values of k for which 2x2 + 2y2 − 6x + 4y + k = 0 represents acircle. 4

Part Marks Level Calc. Content Answer U2 OC44 A CN G9, G15, A6 k <

132 OB 08-006

•1 pd: express circle in std. form•2 ss: know to consider radius•3 pd: process•4 ic: complete

•1 x2 + y2 − 3x + 2y + k2 = 0

•2 g = 32 , f = −1, c = k

2•3 ( 3

2 )2 + 1 − k2 > 0

•4 k <132

[END OF PAPER 1 SECTION B]

hsn.uk.net



Higher Mathematics


Paper 21.[SQA] Find the equation of the tangent at the point (3, 4) on the circle

x2 + y2 + 2x − 4y − 15 = 0. 4


hsn.uk.net



Higher Mathematics


2.[SQA] Triangle ABC has vertices A(2, 2) ,B(12, 2) and C(8, 6) .(a) Write down the equation of l1 ,

the perpendicular bisector ofAB. 1

(b) Find the equation of l2 , theperpendicular bisector of AC. 4

(c) Find the point of intersection oflines l1 and l2 . 1

(d) Hence find the equation of thecircle passing through A, B andC. 2

PSfrag replacements

O x

y

A(2, 2) B(12, 2)

C(8, 6)

Part Marks Level Calc. Content Answer U2 OC4(a) 1 C CN G3, G7 x = 7 2001 P2 Q7(b) 4 C CN G7 3x + 2y = 23(c) 1 C CN G8 (7, 1)

(d) 2 A/B CN G8, G9, G10 (x − 7)2 + (y − 1)2 = 26

•1 ic: state equation of a vertical line

•2 pd: process coord. of a midpoint•3 ss: find gradient of AC•4 ic: state gradient of perpendicular•5 ic: state equation of straight line

•6 pd: find pt of intersection

•7 ss: use standard form of circle equ.•8 ic: find radius and complete

•1 x = 7

•2 midpoint = (5, 4)•3 mAC = 2

3•4 m⊥ = − 3

2•5 y − 4 = − 3

2 (x − 5)

•6 x = 7, y = 1

•7 (x − 7)2 + (y − 1)2

•8 (x − 7)2 + (y − 1)2 = 26

or

•7 x2 + y2 − 14x − 2y + c = 0•8 c = 24

hsn.uk.net



Higher Mathematics


3. (a)[SQA] Find the equation of AB, theperpendicular bisector of the linejoing the points P(−3, 1) andQ(1, 9) . 4

(b) C is the centre of a circle passingthrough P and Q. Given that QC isparallel to the y-axis, determine theequation of the circle. 3

(c) The tangents at P and Q intersect atT.Write down(i) the equation of the tangent at Q

(ii) the coordinates of T. 2

PSfrag replacements

Ox

y

A

B

C

Q(1, 9)

P(−3, 1)

Part Marks Level Calc. Content Answer U2 OC4(a) 4 C CN G7 x + 2y = 9 2000 P2 Q2(b) 3 C CN G10 (x − 1)2 + (y − 4)2 = 25(c) 2 C CN G11, G8 (i) y = 9, (ii) T(−9, 9)

•1 ss: know to use midpoint•2 pd: process gradient of PQ•3 ss: know how to find perp. gradient•4 ic: state equ. of line

•5 ic: interpret “parallel to y-axis”•6 pd: process radius•7 ic: state equ. of circle

•8 ic: interpret diagram•9 ss: know to use equ. of AB

•1 midpoint = (−1, 5)•2 mPQ = 9−1

1−(−1)

•3 m⊥ = − 12

•4 y − 5 = − 12 (x − (−1))

•5 yC = 4 stated or implied by •7

•6 radius = 5 or equiv.stated or implied by •7

•7 (x − 1)2 + (y − 4)2 = 25

•8 y = 9•9 T= (−9, 9)

hsn.uk.net



Higher Mathematics


4.[SQA]



5.[SQA] Find the equation of the circle which has P(−2,−1) and Q(4, 5) as the end pointsof a diameter. 3


hsn.uk.net



Higher Mathematics


6.[SQA] The line y = −1 is a tangent to a circle which passes through (0, 0) and (6, 0) .

Find the equation of this circle. 6


hsn.uk.net



Higher Mathematics


7.[SQA]



hsn.uk.net



Higher Mathematics


8.[SQA]



hsn.uk.net



Higher Mathematics


9.[SQA]



hsn.uk.net



Higher Mathematics


10.[SQA]



hsn.uk.net



Higher Mathematics


11.[SQA]



12.[SQA] Find the equation of the tangent at the point (3, 1) on the circlex2 + y2 − 4x + 6y − 4 = 0. 5


hsn.uk.net



Higher Mathematics


13.[SQA]



hsn.uk.net



Higher Mathematics


14.[SQA]



hsn.uk.net



Higher Mathematics


15.[SQA] Find the possible values of k for which the line x − y = k is a tangent to the circlex2

+ y2= 18. 5


16.[SQA]



17.[SQA] Explain why the equation x2+ y2

+ 2x + 3y + 5 = 0 does not represent a circle. 2


hsn.uk.net



Higher Mathematics


18.[SQA] For what range of values of c does the equation x2+ y2 − 6x + 4y + c = 0 represent

a circle? 3


hsn.uk.net



Higher Mathematics


19.[SQA]



hsn.uk.net



Higher Mathematics


20.[SQA]



hsn.uk.net



Higher Mathematics


21.[SQA]



22.[SQA]



hsn.uk.net



Higher Mathematics


23.[SQA]



hsn.uk.net



Higher Mathematics


24.[SQA]



hsn.uk.net



Higher Mathematics


25.[SQA]



hsn.uk.net



Higher Mathematics


26.[SQA]



hsn.uk.net



Higher Mathematics


27.[SQA]



28. The circle with equation x2 + y2 − 10x + 9 = 0 is shown below.

PSfrag replacements

O x

y

A

x2 + y2 − 10x + 9 = 0

The point A lies on the circumference of the circle and the x -axis.

Find the equation of the tangent to the circle at A. 5

Part Marks Level Calc. Content Answer U2 OC45 C CN G11 x = 9 OB 08-005

•1 ss: start to find A•2 pd: complete for A•3 ic: interpret centre•4 ic: interpret gradients•5 ic: state equation of tangent

•1 y = 0 ⇒ x2 − 10x + 9 = 0•2 A(9, 0)•3 centre (5, 0)•4 radius is horizontal, so tangent is

vertical•5 x = 9

hsn.uk.net



Higher Mathematics


29. A garden is being designed so that there are threeareas for planting around a triangular deckedarea.A diagram of the garden, relative to a set ofcoordinate axes is shown to the right.The points P, Q and R lie on the circle withequation x2 + y2 = 4. The triangle PQR isequilateral, and the line PR is parallel to thex -axis.(a) Find the equation of the line through P and

Q. 4

(b) (i) Find the coordinates of the point P.(ii) Hence calculate the shaded area. 7

PSfrag replacements

O x

y

P

Q

R

x2 + y2 = 4

Part Marks Level Calc. Content Answer U2 OC4(a) 4 B CN G3, G2 y =

√3x + 2 OB 08-002

(bi) 4 C CN G12 P(−√

3,−1)

(bii) 3 B CN G1 4π − 3√

3

•1 ss: find Q•2 ic: know that angle QPR = 60◦

•3 ss: use “m = tan θ” for gradient•4 ic: complete equation

•5 ss: substitute linear into circle•6 pd: express in standard form•7 pd: process quadratic eqn.•8 ic: state coords of P

•9 ss: know to find base•10 pd: find height•11 ic: calculate area

•1 Q(0, 2)•2 angle QPR = 60◦•3 m = tan 60◦ =

√3

•4 y =√

3x + 2

•5 x2 + (√

3x + 2)2 = 4•6 4x2 + 4

√3x = 0

•7 xP = −√

3•8 P(−

√3,−1)

•9 R(√

3,−1) so base PR = 2√

3•10 height = 3•11

π × 22 − 12 × 2

√3 × 3 = 4π − 3

√3

hsn.uk.net



Higher Mathematics


30. The line with equation y + x = 5 meets the circle with equationx2 + y2 − 8x + 2y − 3 = 0 at the points P and Q.

(a) Find the coordinates of P and Q. 4

(b) Find the equation of the circle which has PQ as its diameter. 3

Part Marks Level Calc. Content Answer U2 OC4(a) 4 C CN G12 P(2, 3), Q(8,−3) Ex 2-4-5(b) 3 C CN G9, G10, G6 (x − 5)2 + y2 = 18

•1 ss: substitute linear into quadratic•2 pd: express in standard form•3 pd: process quadratic•4 ic: complete coordinates

•5 ss: find centre•6 ss: find radius•7 ic: state equation of circle

•1 x2 + (5 − x)2 − 8x + 2(5 − x) − 3 = 0•2 2x2 − 20x + 32 = 0•3 x = 2, 8•4 P(2, 3), Q(8,−3)

•5 midpointPQ = (5, 0)

•6 radius: 12 dPQ =

√18

•7 (x − 5)2 + y2 = 18

31. The circle with equation (x − 8)2 + (y − 4)2 = 16is shown in the diagram.The line OA is a non-horizontal tangent to thecircle.(a) Given that OA has gradient m , write down

the equation of OA. 1

(b) Hence or otherwise determine the valueof m . 7

PSfrag replacements

Ox

y

A

Part Marks Level Calc. Content Answer U2 OC4(a) 1 C CN G3 y = mx AT101(b) 7 B CR G12, G13 m = 4/3

•1 ic: state equation of line

•2 ss: know to substitute•3 pd: start to process•4 pd: write in standard form•5 ss: know to use “discriminant = 0”•6 pd: process•7 pd: solve quadratic equation•8 ic: interpret solution

•1 y = mx

•2 (x − 8)2 + (mx − 4)2 = 16•3 x2 − 16x + 64 + · · ·•4 (1 + m2)x2 − (16 + 8m)x + 64 = 0•5 (16 + 8m)2 − 4(1 + m2) × 64 = 0•6 4m − 3m2 = 0•7 m = 0 or m = 4/3•8 non-horizontal so m = 4/3

hsn.uk.net



Higher Mathematics


32. Find the values of k for which the line x + y = k is a tangent to thecircle x2 + y2 − 4x + 2 = 0. 5

Part Marks Level Calc. Content Answer U2 OC45 C CN G13 k = 0, 4 AT031

•1 ss: start to solve simultaneously•2 pd: express in standard form•3 ss: know to use the discriminant•4 ic: substitute values•5 pd: process

•1 x2 + (−x + k)2 − 4x + 2 = 0•2 2x2 − 2(k + 2)x + k2 + 2 = 0•3 for tangency, ∆ = 0•4

∆ = 4(k + 2)2 − 4 × 2(k2 + 2)•5 k = 0, 4

33. The larger circle shown in the diagram hasequation

x2 + y2 − 10x − 4y + 12 = 0.

The smaller circle has centre (10, 134 ) and radius√

174 .

The line through A and B passes through thecentre of both circles.Point A has coordinates (1, 1) .

PSfrag replacements

O x

y

A

B

(a) Show that the circles touch externally. 4

(b) (i) Find the equation of the tangent to the larger circle at A.(ii) Hence state the gradient of the tangent to the smaller circle at B. 4

Part Marks Level Calc. Content Answer U2 OC4(a) 4 A CN G14 proof OB 08-007(bi) 3 C CN G11 5x + y − 6 = 0(bii) 1 C CN G11 −5

•1 ic: interpret centre of circle•2 ss: find radius of circle•3 ss: find sum of radii•4 pd: compare with distance between

centres

•5 ss: find gradient of radius•6 ss: use m1m2 = −1•7 ic: state equation of tangent

•8 ic: interpret gradient

•1 centre (5, 2)•2 radius

√17

•3 sum = 54√

17•4 dcentres = . . . = 5

4√

17 = sum

•5 mrad. = 15

•6 mtgt = −5•7 y − 1 = −5(x − 1)

•8 m = −5

hsn.uk.net



Higher Mathematics


34. Show that the equation x2+ y2

+ kx − 4y − 2 = 0 represents a circle for all valuesof k . 3

Part Marks Level Calc. Content Answer U2 OC43 A CN G9, A6 proof AT018

•1 ss: know to use radius•2 pd: process radius•3 ic: interpret minimum value

•1 need g2+ f 2 − c > 0

•2 k2

4 + 6•3

> 0 for all k

[END OF PAPER 2]

hsn.uk.net



circle - · pdf filethe point p(2,3) lies on the circle (x +1)2 +(y 1)2 = 13. find the...

Documents