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2.1 Analytical Geometry Questions - The Answer

Gr 12 Maths Analytical Geometry Copyright The Answer Work through the Grade 11 Analytical downloads first to ensure your foundation is solid before attempting Grade 12 Analytical Geometry which includes circles. We wish you the best of luck for your exams. From The Answer Series team GR 12 MATHS AANNAALLYYTTIICCAALL GGEEOOMMEETTRRYY Questions and ANSWERS Gr 12 Maths Analytical Geometry : Questions 1 Copyright The Answer GRADE 12 CIRCLES CENTRE AT THE ORIGIN 14. The origin O is the centre of the circle in the figure. P(x; y) and Q(3; -4) are two points on the circle and POQ is a straight line . R is the point (k; 1) and RQ is a tangent to the circle. Determine (leave answers in surd form if necessary): the equation of circle O.

Gr 12 Maths – Analytical Geometry: Questions 1 Copyright © The Answer y S GRADE 12 passes through the origin, write down its equation. (2) B C CIRCLES – CENTRE AT THE ORIGIN 19.2 Determine the gradient of line OP.

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Transcription of 2.1 Analytical Geometry Questions - The Answer

1 Gr 12 Maths Analytical Geometry Copyright The Answer Work through the Grade 11 Analytical downloads first to ensure your foundation is solid before attempting Grade 12 Analytical Geometry which includes circles. We wish you the best of luck for your exams. From The Answer Series team GR 12 MATHS AANNAALLYYTTIICCAALL GGEEOOMMEETTRRYY Questions and ANSWERS Gr 12 Maths Analytical Geometry : Questions 1 Copyright The Answer GRADE 12 CIRCLES CENTRE AT THE ORIGIN 14. The origin O is the centre of the circle in the figure. P(x; y) and Q(3; -4) are two points on the circle and POQ is a straight line . R is the point (k; 1) and RQ is a tangent to the circle. Determine (leave answers in surd form if necessary): the equation of circle O.

2 (2) the length of QT. (3) the length of PQ. (4) the equation of OQ. (2) the coordinates of P. (2) the gradient of QR. (1) the equation of QR in the form y = mx + c. (3) the value of k. (3) whether the point S(3; 2) lies inside, outside or on the circle. Give a reason for your Answer . (2) 15. PRQ is a tangent to the circle with centre O at the point R(6; -2). Calculate the equation of the circle. (2) the tangent. (5) Calculate the size of OQR, rounded off to one decimal digit. (3) Calculate a if (2; a) is a point on the circle x2 + y2 = 40. (3) Determine the coordinates of Q. (2) 16. The straight line y = x - 1 cuts the circle x2 + y2 = 25 at A and B. AB cuts the x-axis at S.

3 The circle intersects the x-axis at T. Calculate the coordinates of A and B. (6) the equation of the circle with centre O and radius OS. (3) the equation of the tangent to the circle at T. (2) 17. In the figure the circle, with centre at the origin O, passes through the point K(- 3; 1). ST is a tangent to the circle at S(1; a). Determine the radius of the circle. (2) Write down the equation of the circle. (1) Show clearly that the value of a = 3. (2) Write down the coordinates of a point Q which is symmetrical to the point S with respect to the x-axis. (2) Write down the gradient of OS. (1) Hence, show that the equation of the tangent ST is given by x + 3y = 10.

4 (3) Write down the coordinates of point T. (2) Hence, prove that TQ is a tangent to the circle. (3) Write down the equation of another tangent to the circle (not shown in the figure above) which is also parallel to ST. (2) 18. In the accompanying figure, the circle with the centre at the origin passes through the point T()-2 3; -2 and cuts the x-axis and y-axis at P and S respectively. Show that the equation of the circle is x2 + y2 = 16 is. (3) Determine the coordinates of P and S. (2) Determine the equation of the line PS. (3) If the straight line QN which is perpendicular to PS passes through the origin, write down its equation. (2) Calculate the coordinates of N if QN meets PS at N.

5 (4) Find the coordinates of a point E, which is the reflection of point T with respect to the line y = -x. (2) 19. A circle with its centre O at the origin passes through the point ()P23;2. Determine the equation of the circle. (2) Determine the gradient of line OP. (1) Hence, without using a calculator, determine the size of the angle between OP and the positive x-axis. (2) Determine the equation of the tangent to the circle at the point ()P23;2 in the form y = ax + q. (4) 20. In the figure, a circle is defined by the equation x2 + y2 = 9, two secants AB and BC meet at a common point B(k; 0), a second line DE passes through the origin and is perpendicular to AB with E a point on the circle. Write down the value of k. (1) Determine the length of AB.

6 (2) Calculate BAC, giving reasons. (2) Calculate the area of ABC. (2) Determine the equation of line DE. (2) Hence, without using a calculator, determine the coordinates of D. (4) Equation: x2+ y2 = r2 STxy A BO x y OS PN()T -2 3; -2 QSee why ABC= 90 in these 2 cases .. A diameter of a circle makes an angle of 90 at the circumference The hypotenuse of a right-angled ABC is the diameter of ?ABC! A tangent to a circle is perpendicular to the radius (or diameter) drawn to the point of contact, BC AB. ACBradius tangentdiameterdiameterACBxyS(1; a)QK(-3; 1) OTR(k; 1)P(x; y) Q(3; -4)Txy O B(k; 0) Ex yACODR(6; -2)QPxyOANALYTICAL GEOMETRYQUESTIONS Gr 12 Maths Analytical Geometry : Questions Copyright The Answer 2 CIRCLES ANY CENTRE 21.

7 Determine the equation of the circle with centre (2; 3) and radius 7 units (3) centre (-1; 4) and radius 5units. (3) Convert the following general form of the equation of a circle to the standard form: x2 - 6x + y2 + 8y - 11 = 0 (3) Write down the centre and the radius of the circle. (3) 23. Determine the equation of the tangent that touches the circle defined by: (x - 1)2 + (y + 2)2 = 25 at the point (4; 2) (4) x2 - 2x + y2 + 4y = 5 at the point (-2; -1) (5) 24. A diameter AB of a circle with points A(-3; -2) and B(1; 4) is given. Determine the equation of the circle. (4) Determine the equation of the tangent to the circle at A. (5) Prove that y = x + 7 is a tangent to the circle x2 + y2 + 8x + 2y + 9 = 0.

8 (6) Determine the point of contact of the tangent and the circle in (2) 26. Find the equation of the circle centre (-2; 5) equal in radius to the circle x2 + y2 + 8x - 2y - 47 = 0. (5) 27. The equation of a circle in the Cartesian plane is x2 + y2 + 6x - 2y - 15 = 0. Rewrite the equation in the form (x - p)2 + (y - q)2 = t. (4) Calculate the length of the tangent drawn to the circle from point P(8; -1) outside the circle. (8) Determine the y-intercepts of the circle. (4) 28. The point M(2; 1) is the midpoint of chord PQ of the circle. x2 + y2 - x - 2y - 5 = 0 Determine the coordinates of the centre, A, of the circle. (2) Determine the radius of the circle.

9 (2) If chord PQ AM, determine the equation of chord PQ. (4) Calculate the coordinates of P and Q. (5) Determine the equation of the tangent to the circle at the point (2; 3). (4) 29. A(3; -5) and B(1; 3) are two points in a Cartesian plane. Calculate the length of AB and leave the Answer in simplified surd from if necessary. (2) Determine the equation of the circle with AB as diameter in the form: (x - a)2 + (y - b)2 = r2 (4) Determine the equation of the tangent to the circle at A in the form: y = mx + c. (5) 30. The equation of a circle with radius 32 units is x2 + y2 - 6x + 2y - m = 0 Determine the coordinates of the centre of the circle.

10 (4) Determine the value of m. (3) 31. A circle with centre M(-4; 2) has the points O(0; 0) and N(-2; y) on the circumference. The tangents at O and N meet at P. Determine: the equation of the circle. (4) the value of y. (2) the equation of OP. (3) the coordinates of P. (7) the specific type of figure represented by POMN. (2) 32. The vertices A and B of ABC lie on the x-axis. The centre of the inscribed circle of ABC is Q(4; 5). The circle touches AC at the point D(0; 8) and BC at the point E(8; 8). Determine: the equation of the circle. (3) the equation of BC. (3) the gradient of AC.


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