Equation of a Circle Flashcards

All 13 cards in this deck

What is the equation of a circle with centre $(0,0)$ and radius $r$?

e.g. radius 5

$x^2 + y^2 = r^2$

e.g. $x^2 + y^2 = 25$

What is the equation of a circle with centre $(a,b)$ and radius $r$?

e.g. centre $(2,-3)$, radius 4

$(x-a)^2 + (y-b)^2 = r^2$

e.g. $(x-2)^2 + (y+3)^2 = 16$

True or false? The circle $(x+7)^2 + (y-4)^2 = 36$ has centre $(7,-4)$ and radius 36.

False. The centre is $(-7,4)$ and the radius is 6, because the right-hand side is $r^2$.

What are the steps to decide whether a point lies inside, on or outside a circle?

e.g. where does $(1,2)$ lie for $x^2 + y^2 = 25$?

1. Substitute the point into the left-hand side: $1^2 + 2^2 = 5$
2. Compare with $r^2$: $5 < 25$
3. Less than $r^2$ = inside, equal = on, greater = outside: inside

What are the steps to find the equation of a circle given the two endpoints of a diameter?

e.g. diameter from $(1,2)$ to $(5,8)$

1. Centre = midpoint of the diameter: $(3,5)$
2. $r^2$ = squared distance from centre to an endpoint: $2^2 + 3^2 = 13$
3. Write it in the standard form: $(x-3)^2 + (y-5)^2 = 13$

What are the steps to find the coordinates where a line meets a circle?

e.g. $x^2 + y^2 = 25$ and $x = 2y - 5$

1. Substitute the line into the circle equation: $(2y-5)^2 + y^2 = 25$
2. Expand and simplify, then solve: $5y^2 - 20y = 0$, so $y = 0$ or $y = 4$
3. Put each value back into the line: $(-5,0)$ and $(3,4)$

How is the gradient of a tangent related to the gradient of the radius drawn to the point of contact?

e.g. radius of gradient $\frac{5}{2}$

It is the negative reciprocal, because the tangent and radius meet at 90°.

e.g. tangent gradient $-\frac{2}{5}$

What are the steps to find the equation of the tangent to a circle at a given point on it?

e.g. tangent to $x^2 + y^2 = 25$ at $(3,4)$

1. Gradient of the radius from the centre to the point: $\frac{4}{3}$
2. Tangent gradient = negative reciprocal: $-\frac{3}{4}$
3. Use $y - y_1 = m(x - x_1)$: $y - 4 = -\frac{3}{4}(x-3)$

What do you know about the two tangents drawn from the same external point to a circle?

They are equal in length.

What is the size of the angle in a semi-circle?

90°. The angle subtended at any point of the circle by a diameter is a right angle.

What are the steps to find the length of a chord from the radius and the distance of the chord from the centre?

e.g. radius 5, chord 3 from the centre

1. The perpendicular from the centre bisects the chord, giving a right-angled triangle with hypotenuse $r$: sides 3, half-chord, 5
2. Pythagoras for half the chord: $\sqrt{5^2 - 3^2} = 4$
3. Double it: chord $= 8$

What are the steps to test whether a line is a tangent to a circle?

e.g. is $y = 5$ a tangent to $x^2 + y^2 = 25$?

1. Substitute the line into the circle equation: $x^2 + 25 = 25$, i.e. $x^2 = 0$
2. Find the discriminant $b^2 - 4ac$ of the quadratic: $0$
3. Zero = tangent, negative = misses, positive = cuts at two points: tangent

True or false? The perpendicular from the centre of a circle to a chord bisects the chord.

True. It meets the chord at its midpoint, so the radius, half-chord and this perpendicular form a right-angled triangle.