Loci that are circles

The locus of the points that are twice as far from $\displaystyle A(0,0)$ as from $\displaystyle B(3,0)$.

The condition is $\displaystyle {PA}={2PB}$, squared $\displaystyle {PA^{2}}={4PB^{2}}$:

Completing the square: $\displaystyle {(x-4)^{2}+y^{2}}={4}$ – a circle with centre $\displaystyle (4,0)$ and radius $\displaystyle 2$.

• When the ratio is $\displaystyle 1:1$ the locus is a line – the perpendicular bisector. With any other ratio $\displaystyle x^{2}$ and $\displaystyle y^{2}$ do not cancel, and the locus is a circle.

The locus of the points $\displaystyle P$ for which $\displaystyle {\angle APB}={90^{\circ}}$, where $\displaystyle A(-2,1)$ and $\displaystyle B(4,1)$.

$\displaystyle PA\perp PB$, so the product of the slopes is $\displaystyle -1$:

• This is the circle with the segment $\displaystyle AB$ as a diameter – as in the theorem: an inscribed angle standing on a diameter is right.
• The points $\displaystyle A$ and $\displaystyle B$ themselves are not in the locus: there is no angle at them, and in the equation the slopes are not defined ($\displaystyle x\neq-2$, $\displaystyle x\neq4$).

The point $\displaystyle Q$ moves on the circle $\displaystyle {x^{2}+y^{2}}={16}$, and $\displaystyle A(6,0)$. What is the locus of the midpoint $\displaystyle M$ of $\displaystyle AQ$?

• $\displaystyle M(x,y)$ is the midpoint of $\displaystyle AQ$, so $\displaystyle {x}={\frac{6+x_Q}{2}}$ and $\displaystyle {y}={\frac{y_Q}{2}}$.
• Express $\displaystyle Q$ through $\displaystyle M$: $\displaystyle {x_Q}={2x-6}$, $\displaystyle {y_Q}={2y}$.
• $\displaystyle Q$ is on the circle: $\displaystyle {(2x-6)^{2}+(2y)^{2}}={16}$. Divide by $\displaystyle 4$: $\displaystyle {(x-3)^{2}+y^{2}}={4}$.

The locus is a circle with centre $\displaystyle (3,0)$, the midpoint between $\displaystyle A$ and the given circle’s centre, and with half the given radius.

$\displaystyle A(-1,0)$ and $\displaystyle B(2,0)$.
Find the locus of the points twice as far from $\displaystyle A$ as from $\displaystyle B$, and write its equation on the board.

$\displaystyle A(-3,0)$ and $\displaystyle B(3,0)$.
Find the locus of the points $\displaystyle P$ for which $\displaystyle {PA^{2}+PB^{2}}={50}$.

$\displaystyle A(1,1)$ and $\displaystyle B(7,-7)$.
Find the locus of the points $\displaystyle P$ for which $\displaystyle {\angle APB}={90^{\circ}}$, and write its equation on the board.

Which points of the circle are not in the locus? Why?

The point $\displaystyle Q$ moves on the circle $\displaystyle {x^{2}+y^{2}}={36}$, and $\displaystyle A(0,8)$.
Find the locus of the midpoint of $\displaystyle AQ$.