Question

If $PB=PY$ for the circle with center at $P,$ find the possible pairs of $(x,y).$

• A. $(1,9)$
• B. $(0,6)$
• C. $(-1,3)$

Answer 1

Answer: (A), (B)

$(0,6)$

The length of perpendicular from the center P to the chord $XZ$ is $PY$
Similarly, PB is the length of perpendicular from centre P to the chord AC.

As PB = PY, both the given chords are of equal length.

Also, the perpendicular to a chord from the center of the circle bisects the chord.

$\therefore$ B and Y are the midpoints of the chords AC and ZX, respectively.

Hence, BC = ZY
$\Rightarrow 6x+4=2y-8$
$\Rightarrow y=3x+6$

For $x=1,y=3\left(1\right)+6=9\to (1,9)$
For $x=0,y=3\left(0\right)+6=6\to (0,6)$

For $x=-1,$
$BC=6x+4=6(-1)+4=-2\text{units}$
This is not possible as distance cannot be negative.

Hence, only options (a) and (b) are correct.