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Wize High School Grade 10 Math Textbook > Analytic Geometry: Geometric Properties

Properties of Circles

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Example: Properties of Circles

A circle has equation x2+y2=25x^2+y^2=25.

a) Show that the points A(−5,0)A(-5,0) and B(3,−4)B(3,-4) lie on this circle.

Substitute point A into the circle equation:
  • Left side of the equation: x2+y2=(−5)2+(0)2=25x^2+y^2=(-5)^2+(0)^2=25
  • Right side of the equation: 2525
  • Since the left side of the equation equals the right sides of the equation, the point A is on the circle

Substitute point B into the circle equation:
  • Left side of the equation: x2+y2=(3)2+(−4)2=25x^2+y^2=(3)^2+(-4)^2=25
  • Right side of the equation: 2525
  • Since the left side of the equation equals the right sides of the equation, the point B is on the circle
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b) A chord is a line segment joining two points on the circumference of a circle. Find the equation of the perpendicular bisector of the chord AB.

The perpendicular bisector of the chord AB will be perpendicular to AB (have a negative reciprocal slope), and bisect AB (passes through the midpoint of AB)

Slope of the perpendicular bisector
mAB=y2−y1x2−x1=(−4)−03−(−5)=−48=−12\begin{array}{rl} m_{AB}=&\dfrac{y_2-y_1}{x_2-x_1}\\[1em] =&\dfrac{(-4)-0}{3-(-5)}\\[1em] =&\dfrac{-4}{8}\\[1em] =&-\dfrac{1}{2} \end{array}

The perpendicular slope is m⊥=2m_\perp=2

Midpoint of the chord AB
MAB=(x1+x22,y1+y22)=((−5)+32,0+(−4)2)=(−1,−2)\begin{array}{rl} M_{AB}=&\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\\[1em] =&\left(\dfrac{(-5)+3}{2},\dfrac{0+(-4)}{2}\right)\\[1em] =&(-1,-2)\\[1em] \end{array}

So, the equation of the perpendicular bisector is y−y1=m(x−x1)y−(−2)=2(x−(−1))y+2=2x+2y=2x\begin{array}{rcl} y-y_1&=&m(x-x_1)\\ y-(-2)&=&2(x-(-1))\\ y+2&=&2x+2\\ y&=&2x \end{array}

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c) Show that the perpendicular bisector found in part b) passes through the origin (0,0).

Substitute the point (0,0)(0,0) into the equation of the perpendicular bisector y=2xy=2x:
  • Left side of the equation: y=0y=0
  • Right side of the equation: 2x=2(0)=02x=2(0)=0
Since the left and right sides equal, the perpendicular bisector does pass through the origin.

Practice: Properties of Circles

A circle has equation x2+y2=25x^2+y^2=25. The points P(0,5), Q(−5,0), R(−3,4), S(−4,−3)P(0,5),~Q(-5,0),~R(-3,4),~S(-4,-3) all lie on this circle. Find the point of intersection of the chords PQ and RS.