Equation of a Circle Flashcards

All 11 cards in this deck

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

e.g. radius $9$

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

e.g. $x^2 + y^2 = 81$

True or false? The circle with equation $x^2 + y^2 = 36$ has radius $36$.

False. The number on the right is $r^2$, so the radius is $\sqrt{36} = 6$.

What should you draw on a grid for the graph of $x^2 + y^2 = 12.25$?

A circle with centre $(0,0)$ and radius $3.5$.

How do you decide whether a point lies on, inside or outside the circle $x^2 + y^2 = r^2$?

e.g. the point $(3,4)$ and $x^2+y^2=25$

Work out $x^2+y^2$ for the point: less than $r^2$ = inside, equal to $r^2$ = on, greater than $r^2$ = outside.

e.g. $3^2+4^2 = 25 = r^2$, so $(3,4)$ lies on the circle.

What are the steps to find the points where a straight line meets a circle centred at the origin?

e.g. $y = x + 1$ and $x^2 + y^2 = 25$

1. Substitute the line into the circle equation: $x^2 + (x+1)^2 = 25$
2. Expand and rearrange to a 3-term quadratic: $2x^2 + 2x - 24 = 0$
3. Solve for $x$, then substitute back into the line for $y$: $(3,4)$ and $(-4,-3)$

What are the steps to find the equation of the tangent to a circle at a 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. Gradient of tangent = negative reciprocal: $-\frac{3}{4}$
3. Substitute into $y - y_1 = m(x - x_1)$: $y - 4 = -\frac{3}{4}(x-3)$, i.e. $3x + 4y = 25$

True or false? A tangent to a circle touches the circle at exactly one point.

True. A line meeting the circle at one point only is a tangent.

A point $(p,1)$ with $p>0$ lies on the circle $x^2+y^2=4$. How do you find $p$ before working out the tangent?

Substitute $y=1$ into the circle equation: $p^2 = 4-1$, so $p = \sqrt{3}$ (take the positive root).

What is the equation of the tangent to $x^2+y^2=r^2$ at the point $(r,0)$?

e.g. $x^2+y^2=9$ at $(3,0)$

The vertical line $x = r$.

e.g. $x = 3$

Where does the graph of $x^2 + y^2 = r^2$ cross the axes?

e.g. $x^2 + y^2 = 49$

At $(r,0)$, $(-r,0)$, $(0,r)$ and $(0,-r)$.

e.g. $(\pm 7, 0)$ and $(0, \pm 7)$.

True or false? When asked to draw the graph of $x^2 + y^2 = r^2$ on a grid, the circle must be drawn with compasses.

False. A freehand circle is accepted as long as it closely approximates a circle with centre $(0,0)$ and the correct radius.