#### The points A and B have coordinates (2, –3) and (8, 5) respectively, and AB is a chord of a circle with center C, as shown in the diagram.

The point M is the mid-point of AB.

(a) Find an equation for the line through C and M.

Anonymous

0

The point M is the mid-point of AB.

(a) Find an equation for the line through C and M.

Krishna

0

i) The point M is the mid-point of AB.

Step 1: Note down the given points and assign these points with (x_1, y_1) \text{ and } (x_2, y_2).

Step 2:Substitute the values into the midpoint formula.

[FORMULA: The midpoint formula is

M = (\frac{x_1 + x_2}{2},\frac{ y_1 + y_2}{2})

where M is the midpoint of a line segment with endpoints at (x_1, y_1) \text{ and } (x_2, y_2).

Step 3: Simplify fraction.

ii) An equation for the line through C and M.

Method 1:

Step 1: Note down the given points as well as assign these points with the variables

EXAMPLE: ( 7, 4 ) assign this point with (x_1, y_1)

( 2, 0 ) assign this point with (x_2,y_2)

................etc

Step 2: Calculate gradient or slope(m) from two points

FORMULA: \frac{y-change}{x-change},

or \frac{y_2 - y_1}{x_2 - x_1}

EXAMPLE: Slope of A(7, 4) and B(2, 0 )

\frac{4-0}{7-2} = \frac{4}{5}

Step 3: Find the slope of the another line.

NOTE: i) Parallel lines have the same slope.

ii) Perpendicular lines have slopes that are opposite reciprocals, like

\frac{a}{b} \text{ and } \frac{-b}{a}. The slopes also have a product of -1

Step 4: Substitute either values slope(m) and any one point into the equation of a straight line.

FORMULA: Equation of a straight line

y - y_1 = m(x - x_1)

Where m = slope and ( x_1, y_1) = any point

EXAMPLE: I took (2, 0) as a point and m = \frac{4}{5}

y - 0 = \frac{4}{5} ( x - 2)

5y = 4x - 8

Step 5: Simplify and make the equation in the form of Ax + By + C =0

Method 2:

Step 1: Note down the given points as well as assign these points with the variables

EXAMPLE: ( 7, 4 ) assign this point with (x_1, y_1)

( 2, 0 ) assign this point with (x_2, y_3)

................etc

Step 2: Substitute either given points in the formula.

FORMULA:

(y_1-y_2)x+\left(x_2-x_1)y+(x_1y_2-x_2y_1)\right)

Step 3: Simplify and make the equation in the form of Ax+ By + C =0