ScholarHigher MathematicsHomework Session

Thursday 29th January 7:30pm

You will need a pencil, paper and a calculator for some of the activities

SCHOLAR online tutor for Maths

and

Author of the new SCHOLAR

National 5 & Higher Maths courses

Margaret Ferguson

The Straight Line

and

The Circle

Tonight’s Revision Sessionwill cover

The distance formula is D =The gradient formula is m = A horizontal line has gradient 0The equation of a horizontal straight line which passes through the point (a,b) is y = bA vertical straight line has an undefined gradientThe equation of a vertical straight line which passes through the point (a,b) is x = aThe equation of a straight line with gradient m and y-intercept (0,c) is y = mx + cThe equation of a straight line with gradient m and passes through the point (a,b) is y - b = m(x - a)

The Straight Linethings you show know

y

b

xa

The general form of the equation is Ax + By + C = 0To be able to identify the gradient or y-intercept:

1. the equation of a straight line must be in the form y = mx + c

2. if the equation is not in that form it must be re-arranged

Parallel lines have equal gradients and distinct y-interceptsA set points are Collinear if they all lie on a single straight line:

3. find the gradients of the lines joining pairs of points4. equal gradients indicate parallel lines5. a common point determines collinearity

m = tan θ where θ is the angle between a line and the positive direction of the x-axisLines with gradients m1 & m2 are perpendicular if m1 × m2 = -1

The midpoint formula is

The Straight Linemore things you must know

yWhat is the equation of the line l2 which passes through the point (8,-2) and is perpendicular to the line l1.

1. Find the gradient of the line l1?What is the given angle in degrees?What is the angle between l1 and the positive direction of the x-axis?What is the gradient of the line l1?

45°

135°

ml1 = tan135° = -1

x

l1

??

2. Find the equation of the line l2?ml2 = 1What is the gradient of l2?

Why?What is the equation of the line l2?

for perpendicular lines ml1 x ml2 = -1y + 2 = 1(x – 8)y = x - 10

If the points M(a, b), N(c, d) and P(e, f) are the vertices of a triangle then the coordinates of the centroid areThe centroid lies two thirds of the distance along each median measured from its vertex.

A median of a triangle is a line from a vertex to the midpoint on the opposite sideThe medians of a triangle are concurrent and the point of intersection is called the centroid.

Triangles: A MedianA

C

B

An altitude of a triangle is a line from a vertex, which is perpendicular to the opposite side.

Altitudes and Perpendicular Bisectors

A

C

B

The perpendicular bisector of a side in a triangle is the line which is perpendicular through the midpoint of the side. A B

C

The altitudes of a triangle are concurrent and the point of intersection is called the orthocentre

The perpendicular bisectors of the sides of a triangle are concurrent and the point of intersection is called the circumcentre.

You must know which is which : median, altitude and perpendicular bisector.

(a) What is the equation of the line p, the median from C?What is a Median?What are the coordinates of R, the midpoint of AB?What is the gradient of CR?Look at the coordinates of C and R to identify the equation of p (the median from

C).

(b) What is the equation of the line q, the altitude from A?What is an Altitude?What is the gradient of BC?What is the gradient of the line perpendicular to BC?Now find the equation of the line q (the altitude from A).

(c) What are the coordinates of the point of intersection of the lines p and q?Use simultaneous equations with your answers to the previous parts of this

question.

The Triangle ABC has vertices A(-1,6), B(-3,-2) and C(5,2).

R(-2,2)mCR = 0, CR is a horizontal line

y = 2

mBC = ½ mperp = -2

y - 6 = -2(x + 1)y = -2x + 4

2 = -2x + 4-2 = -2x x = 1

The point of intersection of the lines p and q is (1,2)

✓A -2B -⅔C 1.5D 2

A line l has equation 3y + 2x = 6.What is the gradient of any line parallel to l?

Vote for the correct answer now

Solution:3y + 2x = 63y = -2x + 6

Remember gradients of parallel lines are equal!!

The equation of a circle with centre (0,0) and radius r is

x2 + y2 = r2

The equation of a circle with centre (a, b) and radius r is

(x - a)2 + (y - b)2 = r2

The general equation of the circle is

x2 + y2 + 2gx + 2fy + c = 0with centre = (-g, -f)

and radius =

The Equation of the Circle

The relationship between a line and a circle can be found by1 substituting the equation of the line in the equation of the circle 2 collecting like terms to obtain a quadratic equation3 evaluating and interpreting the discriminant• if b2 – 4ac = 0 then the line is a tangent to the circle• if b2 – 4ac > 0 then there are 2 points of intersection• if b2 – 4ac < 0 then the line does not meet the circle at all

Any points of intersection can be found by solving the quadratic

It is useful to remember that the angle in a semi-circle is a right angle.

Lines & Circles

A

B

C

Circles which do not touch can be:1. one inside the other 2. or completely apart

Circles can touch:3. internally 4. or externally

Circles may intersect

The distance between the centres of circles and the sum of their radii can help to determine the relationship between the circles

The journey between the coordinates of centres or points of contact can help in problem solving situations

Intersecting circles

A Circle has equation x2 + y2 + 8x + 6y – 75 = 0.What is the radius of the circle?Vote for the correct answer now

A 5

B 10

C

D

✓The general equation of the circle takes the form x2 + y2 + 2gx + 2fy + c = 0

2g = 8 so g = 42f = 6 so f = 3

radius =

so r = = √100 = 10

Show that the line with equation y = 3 – x is a tangent to the circle with equation x2 + y2 + 14x + 4y – 19 = 0 and find the coordiantes of the point of contact P. Substitute the equation of the line into x2 + y2 + 14x + 4y – 19 = 0

x2 + (3 – x)2 + 14x + 4(3 – x) – 19 = 0x2 + (9 – 6x + x2) + 14x + (12 – 4x) – 19 = 0

Collect like terms to obtain a quadratic equation2x2 + 4x + 2 = 0

Evaluate and interpret the discriminantb2 – 4ac = 42 – 4(2)(2)

= 16 – 16 = 0

Since b2 – 4ac = 0 the line y = 3 – x is a tangent to the circle.

Find the coordinates of the point of contact 2x2 + 4x + 2 = 02(x2 + 2x + 1) = 0(x +1)(x +1) = 0x = -1

Let x = -1 in the equation of the tangenty = 3 – (-1) = 4

Hence P has coordinates (-1,4).

Relative to a suitable set of coordinate axes, the diagram below shows the circle from the previous question an a smaller circle with centre C.The line y = 3 – x is a common tangent at the point P.

The radius of the larger circle is 3 times that of the smaller circle.

Find the equation of the smaller circle.

The equation of the larger circle isx2 + y2 + 14x + 4y – 19 = 0

so the centre is (-7,-2)and the radius of the larger circle is

The radius of the smaller circle is ⅓ of the larger circle

Use the diagram to find the coordinates of the centre C.

(-7,-2)

(-1,4)

6

6

2 2

(1,6)Double check the distance from P to C.

Hence the equation of the smaller circle is (x - 1)2 + (y – 6)2 = 8

Question Time

• If you have any questions about tonight’s session please

ask

• I can only answer one question at a time so take turns

• The next session will be on 12th February at 7:30pm

• The topics covered will be Integration & Vectors

• Carol will provide a link for you to give us feedback