Flashcards · 11 cards

Coordinate geometry & circles

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Rule
How do you find the perpendicular bisector of $AB$ ?
Answer
The line through the midpoint of $AB$ with gradient $-\dfrac{1}{m_{AB}}$ .
The perpendicular bisectorThrough M, with gradient −1 ÷ (gradient of AB)
From the lesson: Straight lines
Rule
The angle between two lines with gradients $m_1$ and $m_2$ ?
Answer
$\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1 m_2}\right|$ .
The angle between two linestan θ = |(m₁ − m₂) ÷ (1 + m₁m₂)|
From the lesson: Straight lines
Know it
The point dividing $AB$ internally in the ratio $m : n$ ?
Answer
$\left(\dfrac{nx_1 + mx_2}{m + n}, \dfrac{ny_1 + my_2}{m + n}\right)$ .
From the lesson: Straight lines
Rule
The point dividing $AB$ externally in the ratio $m : n$ ?
Answer
$\left(\dfrac{mx_2 - nx_1}{m - n}, \dfrac{my_2 - ny_1}{m - n}\right)$ .
External division((mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n))
From the lesson: Straight lines
Rule
The equation of the circle with centre $(a, b)$ and radius $r$ ?
Answer
$(x - a)^2 + (y - b)^2 = r^2$ .
A circle(x − a)² + (y − b)² = r²
From the lesson: The equation of a circle
Know it
The centre and radius of $x^2 + y^2 + 2gx + 2fy + c = 0$ ?
Answer
Centre $(-g, -f)$ , radius $\sqrt{g^2 + f^2 - c}$ . First make the $x^2$ and $y^2$ coefficients both 1.
From the lesson: The equation of a circle
Rule
The tangent to a circle at a point $P$ ?
Answer
It is perpendicular to the radius at $P$ : gradient $= -1 \div$ (gradient of the radius).
The tangent at Pgradient of tangent = −1 ÷ gradient of radius
From the lesson: The equation of a circle
Rule
How do you find where a line meets a circle?
Answer
Substitute the line into the circle and solve the quadratic. Two roots: it cuts the circle; one repeated root: it is a tangent; none: it misses.
A line cutting a circleSubstitute the line into the circle and solve
From the lesson: The equation of a circle
Know it
How do you find the circle through three given points?
Answer
Put each point into $x^2 + y^2 + 2gx + 2fy + c = 0$ : three equations in $g$ , $f$ and $c$ .
From the lesson: The equation of a circle
Which method?
NECO 2023 · Paper 2 · Q9 (a)
Find the centre and radius of the circle $3x^2 + 3y^2 + 12x - 6y - 45 = 0$ .
What must you do before reading off $g$ and $f$ ?
Answer
Divide by 3 so that $x^2$ and $y^2$ each have coefficient 1: $x^2 + y^2 + 4x - 2y - 15 = 0$ . Then the centre is $(-g, -f) = (-2, 1)$ .
From the lesson: The equation of a circle Try the question
Which method?
WAEC 2019 · Paper 2 · Q9
The curve $y = 7 - \dfrac6x$ and the line $y + 2x - 3 = 0$ intersect at two points. Find the:
coordinates of the two points;
equation of the perpendicular bisector of the line joining the two points.
What do you need for the perpendicular bisector in (b)?
Answer
The two meeting points from (a), their midpoint, and the gradient perpendicular to the line joining them.
From the lesson: Straight lines Try the question