P1 §6. Circles

Pure mathematics Year 1

Here's a short introduction to Pure maths 1 Chapter 6 Circles.

 

6.1 Midpoints and perpendicular bisectors: Find the mid-point of a line segment and the equation of the perpendicular bisector to a line segment.

6.2 Equation of a circle: Learn how to find the equation of a circle.

6.3 Intersections of straight lines and circles: Solve geometric problems involving straight lines and circles.

6.4 Use tangent and chord properties: Use circle properties to solve problems on coordinate grids.

6.5 Circles and triangles: Find the angle in a semicircle and solve other problems involving circles and triangles.

6.6 Mixed exercise for chapter 6

6.7 Review exercise for chapter 6

 

Prior knowledge check

 

Q1. [P1 §2.2 Completing the square] Write each of the following in the form $(x+p)^2+q$ (i.e. complete the square):

(a) $x^2 + 10x + 28$

(b) $x^2 - 6x + 1$

(c) $x^2 - 12x$

(d) $x^2 + 7x$

 

Answers:

(a) $(x+5)^2+3$

(b) $(x-3)^2-8$

(c) $(x-6)^2-36$

(d) $\left(x+\frac72\right)^2-\frac{49}{4}$

 

Q2. [P1 §5.2 Equations of straight lines] Find the equation of the line passing through each of the following pairs of points:

(a) $A(0,-6)$ and $B(4,3)$

(b) $P(7,-5)$ and $Q(-9,3)$

(c) $R(-4,-2)$ and $T(5,10)$

 

Answers:

(a) $ y = \frac{9}{4} x - 6 $

(b) $ y = -\frac12x - \frac32 $

(c) $ y = \frac43x + \frac{10}{3} $

 

Q3. [P1 §2.5 The discriminant] Use the discriminant to determine whether the following have two real solutions, one real solution or no real solutions.

(a) $x^2 - 7x + 14 = 0$

(b) $x^2 + 11x + 8 = 0$

(c) $4x^2 + 12x + 9 = 0$

 

Answers:

(a) $b^2-4ac=-7$: No real solutions.

(b) $b^2-4ac=193$: Two real solutions.

(c) $b^2-4ac=0$: One real solution.

 

Q4. [P1 §5.3 Parallel and perpendicular lines] Find the eqution of the line that passes through the point $(3,-4)$ and is perpendicular to the line with equation $6x-5y-1=0$.

 

Answer: $y=-\frac56x-\frac32$.

 

Comment: Geostationary orbits are circular orbits around the Earth. Meteorologists use geostationary satellites to provide information about the Earth's surface and atmosphere.