Book 6 Chapter 23 Locus

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23
Locus
Case Study
23.1 Concept of Loci
23.2 Equations of Circles
23.3 Intersection of a Straight Line and a Circle
Chapter Summary
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Case Study
On 24 October 2007, Chinas first lunar orbited
Change 1 Satellite was launched from Xichuan.
The satellite orbited the Moon at an altitude of
200 km above the lunar surface.
It carried out a one-year lunar exploration
mission.
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23.1 Concept of Loci
A. Description of Loci
If a point is moving under a specific condition,
then its path is called the locus of the moving
point.
Consider a football rolling on a horizontal and
smooth ground, the locus of the centre of the
football is a straight line as shown in the
figure below.
When a pendulum swings to and fro, the locus of
its tip is an arc of a circle with the length of
the pendulum as the radius.
In mathematics, locus is the set of points that
satisfy or are determined by some specific
conditions. It can be a straight line, curve,
polygon or circle.
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23.1 Concept of Loci
A. Description of Loci
Case 1 A point R moves in a plane such that it
always maintains a fixed distance of 3 cm from a
fixed line L.
Locus The locus of the point R is a pair of
straight lines which are (1) parallel to L
(2) on one of the sides of L respectively
(3) equidistant from L with a distance of 3 cm.
Case 2 A point Q moves in a plane such that it
always maintains an equal distance of 2 cm from
two parallel lines AB and CD.
Locus The locus of the point Q is a straight
line which is (1) parallel to both AB and CD
(2) in the middle of AB and CD
(3) equidistant from AB and CD with a distance
of 2 cm.
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23.1 Concept of Loci
A. Description of Loci
Example 23.1T
Describe the locus of the moving point P under
each of the following conditions. (a) P is
always 3 cm from the origin O. (b) AB is a
straight line and ÐPAB ? ÐPBA.
Solution
(a) A circle centred at O with radius 3 cm.
(b) The perpendicular bisector of AB.
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23.1 Concept of Loci
A. Description of Loci
Example 23.2T
In the figure, l1 and l2 are two parallel lines 6
cm apart. A moving point P is equidistant from l1
and l2. Sketch the locus of P.
Solution
The locus is a line which is parallel to l1
and l2, and is 3 cm from l1 and l2.
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23.1 Concept of Loci
A. Description of Loci
Example 23.3T
In the figure, AB is a straight line. The locus
of a point P is formed by the centres of all the
circles with radius 1 cm and touch AB. Describe
the locus of P.
Solution
A pair of lines parallel to AB and are 1 cm away
from it.
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23.1 Concept of Loci
A. Description of Loci
Example 23.4T
AB is a line segment. A moving point P passes
through A and B and moves in a way such that ÐAPB
? 130. Describe and sketch the locus of P.
Solution
The locus of P is a minor arc APB of a circle.
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23.1 Concept of Loci
B. Describe Loci in Algebraic Equations
Besides verbal description of the locus, we can
also describe the locus of a point in an
algebraic equation.
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23.1 Concept of Loci
B. Describe Loci in Algebraic Equations
Example 23.5T
A moving point P is always units from
. Express the locus of P in algebraic
form.
Solution
Let (x, y) be the coordinates of P.
? The locus of P is 4x2 4y2 4x 8y 3 ?
0.
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23.1 Concept of Loci
B. Describe Loci in Algebraic Equations
Example 23.6T
Consider a point K(5, 0) and a horizontal line y
? 2. P is a moving point such that its distance
from the line is equal to its distance from the
point K. Express the locus of P in the form y ?
ax2 bx c.
Solution
Let (x, y) be the coordinates of P.
Since the distance from P to the line is equal to
the distance from P to the point (x, ?2) on the
line,
distance from P to (x, ?2) ? distance from P to
K(5, 0)
? The locus of P is
.
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23.2 Equations of Circles
A. Circles
In section 23.1, we learnt that the locus of a
moving point that keeps a fixed distance from a
fixed point is a circle.
Note The fixed distance is the radius and the
fixed point is the centre of the circle.
In this section, we put the circle on the
coordinate plane and hence find the equation of
the circle.
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23.2 Equations of Circles
A. Circles
Suppose the centre is C(h, k) and the radius is
r. Let P(x, y) be the moving point on the circle,
then from the distance formula, we have
Taking square on both sides, we have (x ? h)2
(y ? k)2 ? r2.
Thus, the equation of the circle is (x ? h)2
(y ? k)2 ? r2, where centre ? (h, k) and radius
? r.
Note This equation is called the
centre-radius form of the circle.
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23.2 Equations of Circles
B. General Form of Equations of Circles
If we expand the equation of the circle (x ? h)2
(y ? k)2 ? r2, the equation can be expressed in
the form
x2 ? 2hx h2 y2 ? 2ky k2 ? r2
x2 y2 ? 2hx ? 2ky (h2 k2 ? r2) ? 0
i.e., x2 y2 Dx Ey F ? 0, where D ? ?2h,
E ? ?2k and F ? h2 k2 ? r2.
This is called the general form of the equation
of a circle.
Notes 1. The coefficients of x2 and y2 are
both equal to 1. 2. D, E and F can be any real
numbers. 3. The right hand side of the general
form of the equation of a circle is 0.
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23.2 Equations of Circles
B. General Form of Equations of Circles
Example 23.7T
If a circle is centred at (?4, 2) and it passes
through (?2, 2), find the equation of the circle.
Express the answer in the general form.
Solution
The radius of the circle ? ?2 ? (?4) ? 2.
The equation of the circle
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23.2 Equations of Circles
B. General Form of Equations of Circles
Example 23.8T
Given that A(3, 5) and B(?9, 7) are the two end
points of a diameter of a circle. (a) Find the
equation of the circle in general form. (b) Find
the coordinates of the points of intersection of
the circle and the x-axis.
Solution
(a) Centre
(b) When y ? 0, we have
x ? ?4 or ?2
Radius
? The points of intersection of the circle and
the x-axis are (?4, 0) and (?2, 0).
The equation of the circle
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23.2 Equations of Circles
B. General Form of Equations of Circles
Example 23.9T
A circle passes through three points O(0, 0),
A(6, 0) and B(0, ?10). (a) Find the equation of
the circle in the general form. (b) Does the
point I(?1, ?2) lie on the circle?
Solution
(a) Let the equation of the circle be x2 y2
Dx Ey F ? 0 ()
Since the three points O, A and B must satisfy
(),
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23.2 Equations of Circles
B. General Form of Equations of Circles
Example 23.9T
A circle passes through three points O(0, 0),
A(6, 0) and B(0, ?10). (a) Find the equation of
the circle in the general form. (b) Does the
point I(?1, ?2) lie on the circle?
Solution
Substituting (1) into (2) and (3),
From (4), 6D ? ?36 D ? ?6
From (5), 10E ? 100 E ? 10
? The equation of the circle is x2 y2 6x
10y ? 0.
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23.2 Equations of Circles
B. General Form of Equations of Circles
Example 23.9T
A circle passes through three points O(0, 0),
A(6, 0) and B(0, ?10). (a) Find the equation of
the circle in the general form. (b) Does the
point I(?1, ?2) lie on the circle?
Solution
(b) Substituting (?1, ?2) into the equation x2
y2 6x 10y ? 0.
L.H.S. ? (?1)2 (?2)2 ? 6(?1) 10(?2) ?
?9 ? R.H.S.
? (?1, ?2) does not satisfy the equation of the
circle.
? I(?1, ?2) does not lie on the circle.
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23.2 Equations of Circles
B. General Form of Equations of Circles
Example 23.10T
The centre of a circle lies on the straight line
x 2y 1 ? 0. The circle passes through S(1, 3)
and T(?2, 0).   (a) Find the equation of the
circle in the general form.   (b) If PS is a
diameter of the circle, find the coordinates of
P.
Solution
(a) Let R(h, k) be the centre of the circle.
Since the centre lies on x 2y 1 ? 0, we have
h 2k 1 ? 0(1)
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23.2 Equations of Circles
B. General Form of Equations of Circles
Example 23.10T
The centre of a circle lies on the straight line
x 2y 1 ? 0. The circle passes through S(1, 3)
and T(?2, 0).   (a) Find the equation of the
circle in the general form.   (b) If PS is a
diameter of the circle, find the coordinates of P.
Solution
(1) ? (2)
Substituting k ? ?2 into (2), we have
The equation of the circle
? The centre ? (3, ?2)
Radius
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23.2 Equations of Circles
B. General Form of Equations of Circles
Example 23.10T
The centre of a circle lies on the straight line
x 2y 1 ? 0. The circle passes through S(1, 3)
and T(?2, 0).   (a) Find the equation of the
circle in the general form.   (b) If PS is a
diameter of the circle, find the coordinates of P.
Solution
(b) Let (x, y) be the coordinates of P.
Since (3, ?2) is the mid-point of P and S, we
have
? The coordinates of P are (5, ?7).
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23.2 Equations of Circles
C. Features of Equations of Circles
Remarks
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23.2 Equations of Circles
C. Features of Equations of Circles
Example 23.11T
Consider a circle x2 y2 5x 10y 15 ? 0
Determine whether the point A(1, 1) lies on,
inside or outside the circle.
Solution
Centre
Radius
Distance between the point A and the centre
? radius
? A(1, ?1) lies outside the circle.
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23.2 Equations of Circles
C. Features of Equations of Circles
Example 23.12T
The following are the equations of two circles
C1 x2 y2 2x 14y 46 ? 0 C2 x2
y2 12x 10y 164 ? 0 (a) Find the centres
and radii of the two circles. (b) Hence show
that the two circles touch each other internally.
Solution
(a) For C1,
centre
and
radius
For C2,
centre
and
radius
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23.2 Equations of Circles
C. Features of Equations of Circles
Example 23.12T
The following are the equations of two circles
C1 x2 y2 2x 14y 46 ? 0 C2 x2
y2 12x 10y 164 ? 0 (a) Find the centres
and radii of the two circles. (b) Hence show
that the two circles touch each other internally.
Solution
(b) Distance between the centres
Difference of the radii of the two circles ? 15
2 ? 13
? Distance between the centres
? The two circles touch each other internally.
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23.3 Intersection of a Straight Line and
a Circle
In the same coordinate plane, there are three
cases showing the relationship between the graphs
of a circle x2 y2 Dx Ey F ? 0 and a
straight line y ? mx c.
Case 1 intersect at two distinct points Case
2 intersect at one point only Case 3 no
point of intersection.
Notes For case 2, the straight line is called a
tangent to the circle.
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23.3 Intersection of a Straight Line and
a Circle
Without the actual drawing of the graphs, the
number of points of intersection of the two
graphs can be determined algebraically by
carrying out the following steps.
Step 1 Use the method of substitution to
eliminate one of the unknowns (either x or y) of
the simultaneous equations. We can then obtain a
quadratic equation in one unknown.
Substituting (1) into (2), x2 (mx c)2
Dx E(mx c) F ? 0
x2 m2 x2 2mcx c2 Dx Emx
Ec F ? 0
(1 m2 )x2 (2mc D Em)x (c2 Ec F) ? 0
... ()

• Step 2 Evaluate the discriminant (D) of the
  quadratic equation ().
• If D gt 0, there are two points of intersection.
• If D ? 0, there is only one point of
  intersection.
• If D lt 0, there is no point of intersection.

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23.3 Intersection of a Straight Line and
a Circle
Example 23.13T
Find the coordinates of the points of
intersection of the straight line 2x y 1 ? 0
and the circle x2 y2 3x y 10 ? 0.
Solution
Consider the simultaneous equations of the
straight line and the circle.
When x ? ?1, y ? 2(?1) ? 1 ? ?3
Substituting (1) into (2), we have
When x ? 2, y ? 2(2) ? 1 ? 3
The points of intersection are (?1, ?3) and (2,
3).
x ? ?1 or 2
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23.3 Intersection of a Straight Line and
a Circle
Example 23.14T
If the straight line y 3x 1 ? 0 meets the
circle x2 y2 2x 4y k ? 0 at two distinct
points, find the range of values of k.
Solution
Consider the simultaneous equations of the
straight line and the circle.
Substituting (1) into (2),
Since the straight line meets the circle at two
distinct points, the discriminant of () is
greater than 0.
? The range is k lt 5.
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23.3 Intersection of a Straight Line and
a Circle
Example 23.15T
Given a circle (x 1)2 (y 2)2 ? k. If P(4,
3) lies on the circle, find (a) the centre of
the circle, (b) the value of k, (c) the
equation of the tangent to the circle at P.
Solution
(a) Centre
(b) Substituting (4, ?3) into the equation
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23.3 Intersection of a Straight Line and
a Circle
Example 23.15T
Given a circle (x 1)2 (y 2)2 ? k. If P(4,
3) lies on the circle, find (a) the centre of
the circle, (b) the value of k, (c) the
equation of the tangent to the circle at P.
Solution
(c) Slope of the line joining the centre and P
Slope of the tangent
? Equation of the tangent
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Chapter Summary
23.1 Concept of Loci
1. A locus is the set of all points that satisfy
the given specified conditions.
2. The locus of points can be expressed in
algebraic equations.
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Chapter Summary
23.2 Equations of Circles
1. A circle can be expressed by the centre-radius
form (x ? h)2 (y ? k)2 ? r2, where centre ?
(h, k) and radius ? r.
2. The general form of a circle is x2 y2 Dx
Ey F ? 0.
4. If the distance between the centre of a circle
and a point P is (a) smaller than the radius of
the circle, then P lies inside the circle (b)
equal to the radius of the circle, then P lies on
the circle (c) greater than the radius of the
circle, then P lies outside the circle.
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Chapter Summary
23.3 Intersection of a Straight Line and a Circle
1. There are three cases showing the
relationship between a straight line and a
circle (a) Intersect at two points (b) Intersec
t at one point (c) No point of intersection We
can use the discriminant to determine the number
of the points of intersection.
2. The coordinates of the point(s) of
intersection can be found by solving the
simultaneous equations of a straight line and a
circle.
3. If a straight line touches the circle at one
point only, then the straight line is a tangent
to the circle.
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Follow-up 23.1
23.1 Concept of Loci
A. Description of Loci
Describe the locus of the moving point P where
the distance from P to a fixed line L is always 2
units.
Solution
The locus of P is a pair of parallel lines, each
of them is 2 units from the given line L.
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Follow-up 23.2
23.1 Concept of Loci
A. Description of Loci
Describe and sketch the locus of the moving point
P under the given condition. Condition A point
P moves such that it is equidistant from the
lines OM and ON.
Solution
The locus of point P is the angle bisector of
?MON.
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Follow-up 23.3
23.1 Concept of Loci
A. Description of Loci
The figure shows a rectangle with length 6 cm and
width 4 cm. A circle with centre P and radius 1
cm is moving inside the rectangle. Describe the
locus of P if the circle always touches at least
one side of the rectangle.
Solution
A rectangle with length 4 cm and width 2 cm.
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Follow-up 23.4
23.1 Concept of Loci
A. Description of Loci
AB is a line segment and a moving point P passes
through A and B and moves in a way such that ÐAPB
? 80. Describe and sketch the locus of P.
Solution
The locus of P is a major arc APB of a circle.
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Follow-up 23.5
23.1 Concept of Loci
B. Describe Loci in Algebraic Equations
A moving point R is always 10 units from Q(2,
2). Express the locus of R in algebraic
equation.
Solution
Let (x, y) the coordinates of R.
Since the distance between R and Q is
,
? The locus of R is x2 y2 4x 4y 92 ? 0.
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Follow-up 23.6
23.1 Concept of Loci
B. Describe Loci in Algebraic Equations
Consider a point R(0, 3) and a horizontal line y
? 3. P is a moving point such that its distance
from the line is equal to its distance from the
point R. Describe the locus of P in the form y ?
ax2 bx c.
Solution
Let (x, y) be the coordinates of P.
Since the distance from P to the line is equal to
the distance from P to the point (x, 3) on the
line, distance from P to (x, 3) ? distance from
P to R(0, ?3)
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Follow-up 23.7
23.2 Equations of Circles
B. General Form of Equations of Circles
If a circle is centred at (3, 0) and it passes
through (3, 6), find the equation of the circle.
Express the answer in the general form.
Solution
The radius of the circle
The equation of the circle
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Follow-up 23.8
23.2 Equations of Circles
B. General Form of Equations of Circles
Given that A(3, 4) and B(7, 10) are the two end
points of a diameter of a circle. (a) Find the
equation of the circle in the general form. (b)
If P(5, k) is a point on the circle, find the
values of k.
Solution
(a) Centre
Radius
The equation of the circle
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Follow-up 23.8
23.2 Equations of Circles
B. General Form of Equations of Circles
Given that A(3, 4) and B(7, 10) are the two end
points of a diameter of a circle. (a) Find the
equation of the circle in the general form. (b)
If P(5, k) is a point on the circle, find the
values of k.
Solution
(b) When x ? 5, we have
y ? 2 or 12
? The values of k are 2 or 12.
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Follow-up 23.9
23.2 Equations of Circles
B. General Form of Equations of Circles
Suppose a circle passes through three points O(0,
0), A(3, 0) and B(0, 3). (a) Find the equation
of the circle in the general form. (b) Does the
point P(3, 3) lie on the circle?
Solution
(a) Let the equation of the circle be x2 y2
Dx Ey F ? 0 ()
Since the three points O, A and B must satisfy
(),
Substituting (1) into (2) and (3),
From (4), 3D ? ?9
D ? ?3
From (5), 3E ? 9
E ? 3
? The equation of the circle is x2 y2 3x
3y ? 0.
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Follow-up 23.9
23.2 Equations of Circles
B. General Form of Equations of Circles
Suppose a circle passes through three points O(0,
0), A(3, 0) and B(0, 3). (a) Find the equation
of the circle in the general form. (b) Does the
point P(3, 3) lie on the circle?
Solution
(b) Substituting (3, ?3) into the equation
x2 y2 3x 3y ? 0.
L.H.S. ? 32 (3)2 3(3) 3(3)
? 0
? R.H.S.
? (3, ?3) satisfies the equation of the circle.
? P(3, ?3) lies on the circle.
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Follow-up 23.10
23.2 Equations of Circles
B. General Form of Equations of Circles
The centre of a circle lies on the straight line
x y 2 ? 0. The circle passes through P(1, 4)
and Q(5, 4). (a) Find the centre and the radius
of the circle. (b) Hence find the equation of
the circle in the general form.
Solution
(a) Let R(h, k) be the centre of the circle.
Since the centre lies on x y ? 2 ? 0,
we have h k 2 ? 0.(1)
Since PR ? QR ? radius, we have
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Follow-up 23.10
23.2 Equations of Circles
B. General Form of Equations of Circles
The centre of a circle lies on the straight line
x y 2 ? 0. The circle passes through P(1, 4)
and Q(5, 4). (a) Find the centre and the radius
of the circle. (b) Hence find the equation of
the circle in the general form.
Solution
(1) ? 3 ? (2)
Substituting k ? 0 into (1), h ? 2
? The centre
Radius
(b) The equation of the circle
49
Follow-up 23.11
23.2 Equations of Circles
C. Features of Equations of Circles
Consider a circle 4x2 4y2 ? 10. (a) Find the
equation of the circle in the general form. (b)
Determine whether the point
lies on, inside or outside the circle.
Solution
(a)
(b) Centre ? (0, 0)
Radius
Distance between T and the centre
? radius
? T lies on the circle.
50
Follow-up 23.12
23.2 Equations of Circles
C. Features of Equations of Circles
The following are the equations of two circles
C1 x2 y2 8x 6y 21 ? 0 and C2 x2 y2
12y 27 ? 0 (a) Find the centres and radii of
the two circles. (b) Hence show that the two
circles touch each other.
Solution
(b) Distance between the centres
(a) For C1
Centre
Radius
Sum of the radii ? 2 3 ? 5
For C2
Centre
? Distance between the centres
Radius
? The two circles touch each other externally.
51
Follow-up 23.13
23.3 Intersection of a Straight Line and
a Circle
Find the coordinates of the points of
intersection of the straight line x y 2 ? 0
and the circle x2 y2 x 3y 4 ? 0.
Solution
Consider the simultaneous equations of the
straight line and the circle.
From (1),
When y ? 1, x ? 3.
Substituting (3) into (2), we have
When y ? ?1, x ? 1.
The points of intersection are (3, 1) and (1,
?1).
y ? 1 or ?1
52
Follow-up 23.14
23.3 Intersection of a Straight Line and
a Circle
If the straight line 2y x 3 ? 0 meets the
circle x2 y2 x 10y k ? 0 at only one
point, find the value of k.
Solution
Consider the simultaneous equations of the
straight line and the circle.
Substituting (1) into (2),
Since the straight line meets the circle at only
one point, the discriminant of () is equal to 0.
53
Follow-up 23.15
23.3 Intersection of a Straight Line and
a Circle
Given a circle x2 y2 10x 6y 17 ? 0. (a)
Find the centre of the circle. (b) If P(1, 4)
is a point on the circle, find the equation of
the tangent to the circle at P.
Solution
(a) Centre
(b) Slope of the line joining centre to P
Slope of the tangent
? Equation of the tangent