State whether the following statement is true or false.
Enter $1$ for true and $0$ for false
A circle has its center at the origin and a point $P (5, 0)$ lies on it. The point $Q (6, 8)$ lies outside the circle.

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Solution

$A p o i n t Q (x, y) w i l l l i e o u t s i d e a c i r c l e w i t h i t s c e n t r e a t t h e o r i g i n e$

$i f t h e l e n g t h o f i t s r a d i u s r i s l e s s t h a n t h e d i s t a n c e o f t h e p o i n t$

$Q (x, y) f r o m t h e o r i g i n e .$

$H e r e t h e c o - o r d i n a t e s o f t h e p o i n t Q i s (6, 8) .$

$∴ T h e d i s t a n c e d o f t h e p o i n t Q (x, y) f r o m t h e o r i g i n e i s$

$d = \sqrt{x^{2} + y^{2}} = \sqrt{6^{2} + 8^{2}} u n i t s = 10 u n i t s .$

$A g a i n t h e r a d i u s r = d i s t a n c e o f P (5, 0) f r o m t h e o r i g i n e .$

$(s i n c e P l i e o n t h e c i r c l e)$

$⟹ r = \sqrt{5^{2} + 0^{2}} u n i t s = 5 u n i t s .$

$∴ d > r$

$S o t h e p o i n t Q l i e s o u t s i d e t h e c i r c l e .$

$i . e T h e s t a t e m e n t i s t r u e$

$A n s - T R U E .$

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