Answer to Question #316374 in Analytic Geometry for Chloe

Question

A line segment has end points A (-12, -11) and B (6, 19). Find the three points that divide AB into four equal pieces. A) show that the line segment with end points A (5,3) and B (5 ,-5) is a chord of the circle X to the power of 2 + Y to the power of two = 34

Answer 1

To determine the three points that divide AB into 4 equal parts, we shall find a point X on AB such that X is the midpoint of AB. Then we shall find a point Y on AX such that Y is the midpoint of AX. Finally, we shall find a point Z such that Z is the midpoint of XB

Midpoint of AB $=$ $(x,y)$ $=$ $(\frac{-12+6}{2}, \frac{-11+19}{2})$

Midpoint of AB $= (x,y) = (-3, 4)$

$X(-3,4)$

Midpoint of AX $= (x,y) = (\frac{-12-3}{2}, \frac{-11+4}{2})$

Midpoint of AX $= (x,y) = (\frac{-15}{2}, \frac{-7}{2})$

$Y(\frac{-15}{2}, \frac{-7}{2})$

Midpoint of BX $= (x,y) = (\frac{-3+6}{2}, \frac{4+19}{2})$

Midpoint of BX $= (x,y) = (\frac{3}{2}, \frac{23}{2})$

$Z(\frac{3}{2}, \frac{23}{2})$

Equation of circle : $x²+y²=34$

To show that line AB is a chord of the circle, we shall show that one of the points lie on the circle, and the midpoint of line AB is not the centre of the circle

Let $x=-5, y=3$

$x²+y²=(-5)²+3²=25+9=34$

Hence, $A(-5,3)$ lies on the circle

Midpoint of AB $= (x,y) = (\frac{5+5}{2}, \frac{-5+3}{2})$

Midpoint of AB $= (x,y) = (5, -1)$

From $x²+y²=34$ , it is evident that the centre of the circle is $(0,0)$

$(0,0) ≠ (5, -1)$

Hence, line AB is a chord of the circle, $x²+y²=34$