Given the points A (1, -8), B (-4, 7) and C(-7, -4), show that they lie on a circle K Class 9

Given the points A (1, -8), B (-4, 7) and C(-7, -4), show that they lie on a circle K Class 9

Question 1.

i. Given the points A (1, -8), B (-4, 7) and C(-7, -4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle

K?

ii. Given the points D (-5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K.

Solution:

i. $\begin{aligned} \mathrm{OA} & =\sqrt{(1-0)^2+(-8-0)^2} \\ & =\sqrt{1+64}=\sqrt{65} \text { units }\end{aligned}$

$\begin{aligned} \mathrm{OB} & =\sqrt{(-4-0)^2+(7-0)^2}=\sqrt{16+49}=\sqrt{65} \text { units } \\ \mathrm{OC} & =\sqrt{(-7-0)^2+(-4-0)^2} \\ & =\sqrt{49+16}=\sqrt{65} \text { units }\end{aligned}$

∴ OA = OB = OC

∴ Points A, B and C lie on circle K.

Radius of circle = [latex]\sqrt{65}[/latex] units

ii. OD = [latex]\sqrt{(-5-0)^2+(6-0)^2}[/latex]

= [latex]\sqrt{25+36}[/latex] = [latex]\sqrt{61}[/latex] < [latex]\sqrt{65}[/latex]

∴ Point D lies inside the circle.

OE = $\sqrt{(0-0)^2+(9-0)^2}=\sqrt{0+81}=9>\sqrt{65}$

∴ Point E lies outside the circle.