Teaching Example 19.1
The figure shows a line segment AB of length 4 cm. A moving point P maintains a fixed distance of 2 cm from the nearest point on AB. Sketch and describe the locus of P.
Solution
The locus of point P is made up of:
(i)two parallel line segments, each of length equal to AB, and at a distance of 2 cm from AB, and;
(ii)two semi-circles each of radius 2 cm, with centres A and B respectively.
Teaching Example 19.2
The figure shows a square ABCD of side 2 cm. Sketch and describe the locus of a moving point P which is always equidistant from AB and AD, and lies inside or on square ABCD.
Solution
The locus of P is the diagonal AC of square ABCD.
Teaching Example 19.3
The figure shows a square ABCD of side 4 cm, and a circle of radius 1 cm with centre P. If the circle rolls inside the square along the sides of the square, sketch and describe the locus of the centre P of the circle.
Solution
The locus of P is a square of side 2 cm inside square ABCD.
Teaching Example 19.3 (Extra)
In the figure, P is a point on a wheel of radius 1 cm. Initially, the wheel touches the horizontal ground at point P. It then rotates clockwise and rolls to the right. Sketch the locus of point P.
Solution
Teaching Example 19.4
A moving point P maintains a distance of 5 units from a fixed point A(5, 5).
(a)Sketch and describe the locus of P.
(b)Find the equation of the locus of P.
Solution
(a)
The required locus is a circle with centre A(5, 5) and radius 5 units.
(b)Let (x, y) be the coordinates of P.
∴The equation of the locus of P is .
Teaching Example 19.5
A moving point P is equidistant from A(3, 0) and B(1, –4).
(a)Sketch and describe the locus of P.
(b)Find the equation of the locus of P.
Solution
(a)
The required locus is the perpendicular bisector of AB.
(b)Let (x, y) be the coordinates of P.
∴The equation of the locus of P is .
Teaching Example 19.6
A moving point P is equidistant from the point A(–1, 0) and the straight line x = 2. Find the equation of the locus of P.
Solution
Let (x, y) be the coordinates of P, and G be a point on L: x = 2 such that PGL.
∴The equation of the locus of P is .
Teaching Example 19.7
A moving point P maintains an equal distance from two parallel lines and . Find the equation of the locus of P.
Solution
The required locus is the straight line L3 shown below.
∵y-intercept of L1 = 4 and y-intercept of L2 = –6
∴y-intercept of L3 =
Slope of L3 = slope of L1 = 3
∴The equation of L3 is
∴The equation of the locus of P is .
Teaching Example19.7 (Extra)
A moving point P maintains a fixed distance of units from the straight lines . Find the equations of the locus of P.
Solution
The required locus is a pair of straight lines parallel to and at a distance units from L, namely L1 and L2.
With the notations in the figure,
∵Slope of L = 1
∴∠AOC = 45°
∴
∴The y-intercept of L1 is 2.
Similarly,
∴The y-intercept of L2 is –2.
Slope of L1 = slope of L2 = slope of L = 1
The equation of L1 is
The equation of L2 is
∴The required equations are and .
Teaching Example 19.8
Find the equation of each of the following circles with the given centre C and radius r. (Leave your answers in the standard form.)
(a)C(0, 0), r = 4
(b)C(–2, 1), r =
Solution
(a)The equation of the circle is
i.e.
(b)The equation of the circle is
i.e.
Teaching Example 19.9
For each of the following equations of circles, find the centre and the radius of the circle. (Leave your answers in surd form if necessary.)
(a)
(b)
(c)
Solution
(a)
∴centre =, radius =
(b)
∴centre =, radius =
(c)
∴centre =, radius =
Teaching Example 19.10
In the figure, a circle with centre C(–5, 0) passes through the point A(0, 3).
(a)Find the equation of the circle.
(b)Determine whether the point B(1, 4) lies inside, outside or on the circle.
Solution
(a)Radius of the circle
∴The equation of the circle is
(b)
i.e.CB > radius
∴The point B(1, 4) lies outside the circle.
Teaching Example 19.11
For each of the following equations of circles, find the centre and the radius of the circle. (Leave your answers in surd form if necessary.)
(a)
(b)
Solution
(a)Centre
Radius
(b)
Centre
Radius
Teaching Example 19.12
In the figure, A(–2, 7) and B(–7, –5) are the end points of a diameter of a circle, and C is the centre of the circle. Find the equation of the circle in the general form.
Solution
∵C is the mid-point of AB.
∴Coordinates of C
Radius
∴The equation of the circle is
Teaching Example 19.12 (Extra)
It is given that A, B(4, 2) and C(8, –2) are three points on a circle. M is a point inside the circle such that AM = BM = CM.
(a)Determine whether M is the centre of the circle. Explain your answer.
(b)Given that BMC is a straight line, find the equation of the circle passing through A, B and C.
Solution
(a)Yes. Since AM = BM, BM =CM and CM = AM, M is the intersection of the perpendicular bisectors of AB, BC and CA. Therefore, M is the circumcentre of △ABC, which is the centre of the circle.
(b)∵M is the mid-point of BC.
∴Coordinates of M
Radius
∴The equation of the circle is
Teaching Example 19.13
Given a circle which passes through (0, 0), (6, 0) and (3, 1), find the equation of the circle.
Solution
Let be the equation of the circle.
∵The circle passes through (0, 0), (6, 0) and (3, 1).
∴By substitution, we have
By substituting (1) into (2), we have
By substituting (1) and (4) into (3), we have
∴The equation of the circle is .
Teaching Example 19.14
In the figure, C is the centre of the circle, which touches the x-axis at A(–6, 0) and passes through B(6, 8).
(a)Find the coordinates of C.
(b)Find the equation of the circle.
Solution
(a)Let (h, k) be the coordinates of C.
∵The circle touches the x-axis at A(–6, 0).
∴CAx-axis
∴
∵(radii)
∴
∴Coordinates of C
(b)The equation of the circle is
Teaching Example 19.14 (Extra)
A circle touches the y-axis and cuts the x-axis at A(1, 0) and B(9, 0). Find the equation of the circle.
Solution
Let (h, k) be the coordinates of the centre C of the circle.
∵The circle touches the y-axis and cuts the x-axis at A(1, 0) and B(9, 0).
∴x-coordinate of C = x-coordinate of the mid-point M of AB
∴The radius of the circle is 5.
∵AM = MB
∴CMAB(line joining centre to mid-pt. of chord chord)
In △CAM,
(Pyth. theorem)
∴Coordinates of C = (5, 3)
The equation of the circle is
Teaching Example 19.15
In the figure, the circle passes through the origin, and cuts the y-axis at A(0, 4). Its centre C lies on the straight line .
(a)Find the equation of the circle.
(b)The circle cuts the negative x-axis at P. Find the coordinates of P.
Solution
(a)Let O be the origin and M be the mid-point of OA. Let (h, k) be the coordinates of C.
∵CMOA
∴y-coordinate of C = y-coordinate of M
By substituting (h, 2) into , we have
∴Radius
∴The equation of the circle is
(b)By substituting y = 0 into (1), we have
∴Coordinates of P
Teaching Example 19.16
In each of the following, determine the number of intersections between the given straight line L and circle S.
(a),
(b),
Solution
(a)
By substituting (1) into (2), we have
∴There is only one intersection between the straight line and the circle.
(b)
By substituting (3) into (4), we have
∴There are two intersections between the straight line and the circle.
Teaching Example 19.17
In each of the following, find the coordinates of the intersections between the given straight line L and circle S.
(a),
(b),
Solution
(a)
By substituting (1) into (2), we have
By substituting into (1), we have
∴The coordinates of the intersection between the straight line and the circle are (2, 1).
(b)
From (3), we have
By substituting (5) into (4), we have
By substituting into (5), we have
By substituting into (5), we have
∴The coordinates of the intersections between the straight line and the circle are (–1, 7) and (1, 5).
Teaching Example 19.18
If the straight line touches the circle at only one point, find the values of c.
Solution
By substituting (1) into (2), we have
∵The straight line touches the circle at only one point.
∴
Teaching Example 19.19
Find the equation of the tangent to the circle at the point A(2, –1).
Solution
Let C be the centre of the circle.
Coordinates of C
Slope of CA
∴Slope of the tangent at A
∴The equation of the tangent to the circle at A is
Teaching Example 19.20
(a)Find the equation of the straight line passing through A(0, 9) and with slope m. (Express your answer in terms of m.)
(b)Hence, find the equations of the two tangents to the circle from the point A(0, 9).
Solution
(a)∵The straight line passes through A(0, 9).
∴Its y-intercept = 9
The equation of the straight line is
(b)From the result of (a), the equations of the required tangents can be written in the form . …… (1)
By substituting (1) into , we have
∵The straight line is a tangent to the circle.
∴
∴The equations of the two tangents are
i.e.
Teaching Example 19.21
In the figure, is a tangent to the circle .
(a)Find the value of k.
(b)Find the equation of another tangent to the circle S which is parallel to L.
Solution
(a)
From (1), we have
By substituting (3) into (2), we have
∵L is a tangent to the circle.
∴
(b)Slope of L = 1
∵The required tangent is parallel to L.
∴Slope of the tangent = 1
Let c be the y-intercept of the tangent. Then, the equation of the required tangent can be written in the form . …… (3)
By substituting (3) into , we have
∵The straight line is a tangent to the circle.
∴
∴The required equation of the tangent is
Teaching Example 19.21 (Extra)
The circle is inscribed in △ABC. It is given that the equation of AB is , AB BC and AC is a horizontal line segment.
(a)Find the centre and radius of S.
(b)Find the equations of BC and AC.
(c)Find the coordinates of A, B and C.
(d)Find the area of △ABC.
Solution
(a)Coordinates of the centre of S
Radius of S
(b)Since AC is a horizontal line segment,
the equation of AC is y = 4 + 3, i.e. y = 7.
Slope of AB
∵AB BC
∴Slope of BC
Let c be the y-intercept of BC. Then, the equation of BC can be written in the form . …… (1)
By substituting (1) into , we have
∵The straight line is a tangent to the circle.
∴
∴The equation of BC is
(c)By substituting into , we have
∴Coordinates of A
By substituting into , we have
∴Coordinates of C
(2) 3 + (3) 4:
By substituting into (2), we have
∴Coordinates of B
(d)
∴Area of △ABC
1