FINDING THE EQUATION OF RIGHT BISECTOR OF CHORD OF CIRCLE

Problem 1 :

The circle defined by x² + y² = 13

a) verify algebraically all the points M(-3, 2) and N (2, -3) are on the circle.

b) Find the equation in the form of y = mx + b for the right bisector of chord MN.

Solution :

Check if the points M(-3, 2) and N (2, -3) lies on the circle.

So, the points M and N lies on the circle.

Since M and N are on the points of circle, finding the midpoint of the chord, we get

Midpoint = (-3 + 2) / 2, (2 + (-3))/2

M = (-1/2, -1/2)

Slope of the chord = (y₂ - y₁)/ (x₂ - x₁)

M(-3, 2) and N (2, -3)

= (-3 - 2)/(2 + 3)

= -5/5

= -1

Slope of the required line = 1

Problem 2 :

Graph the circle defined by

x² + y² = 45

a) verify algebraically that the line segment joining P(-3, 6) and Q(6, -3) is the chord of the circle. 

b) Find the equation in the form y = ax + b for the right bisector of chord PQ.

Solution :

Connecting any two points on the circle which lies on circumference of the circle is known as chord.

Check if P(-3, 6) lies on the circle :

So, the points P and Q lies on the circle.

Since P and Q are on the points of circle, finding the midpoint of the chord, we get

P(-3, 6) Q(6, -3)

Midpoint = (-3 + 6) / 2, (6 + (-3))/2

M = (3/2, 3/2)

Slope of the chord = (y₂ - y₁)/ (x₂ - x₁)

= (-3 - 6)/(6 + 3)

= -9/9

= -1

Slope of the required line = 1

Problem 3 :

The line with equation y = 1−x intersects the circle with equation

x² + y² + 6x + 2y = 27

at the points A and B. Find the length of the chord AB, giving your answer in the form 2√k .

Solution :

x² + y² + 6x + 2y = 27 ----(1)

y = 1 − x----(2)

Applying the value of y in (1), we get

x² + (1 − x)² + 6x + 2(1 - x) = 27

x² + 1 + x² - 2x + 6x + 2 - 2x = 27

2x²+ 2x + 3 - 27 = 0

2x²+ 2x - 24 = 0

x²+ x - 12 = 0

(x + 4) (x - 3) = 0

x = -4 and x = 3

Applying the value of x in (2), we get

When x = -4, y = 1 + 4 ==> 5

When x = 3, y = 1 - 3 ==> -2

So, the points of intersections are A (-4, 5) and B (3, -2).

Length of chord AB :

= √(x₂ - x₁)² + (y₂ - y₁)²

= √(3+4)² + (-2-5)²

= √7² + 7²

= √98

= √2 x 7 x 7

= 7√2

So, the value of k is 7.