First slide

Equation of line in Straight lines

Question

$I f t h e c i r c u m c e n t r e o f a n a c u t e a n g l e d t r i a n g l e l i e s a t t h e o r i g i n a n d t h e c e n t r o i d i s t h e m i d d l e p o i n t o f t h e l i n e j o i n i n g t h e p o i n t s (a^{2} + 1, a^{2} + 1) a n d (2 a, - 2 a) t h e n t h e o r t h o c e n t r e i s$

Moderate

A

$(\frac{2 {(a - 1)}^{2}}{3}, \frac{2 {(a + 1)}^{2}}{3})$
B

$(\frac{2 {(a + 1)}^{2}}{3}, \frac{2 {(a - 1)}^{2}}{3})$
C

$(\frac{3 {(a + 1)}^{2}}{2}, \frac{3 {(a + 1)}^{2}}{2})$
D

$(\frac{3 {(a + 1)}^{2}}{2}, \frac{3 {(a - 1)}^{2}}{2})$

Solution

$G i v e n c i r c u m c e n t r e S i s (0, 0) a n d t h e c e n t r o i d i s G (\frac{a^{2} + 2 a + 1}{2}, \frac{a^{2} - 2 a + 1}{2}) = (\frac{{(a + 1)}^{2}}{2}, \frac{{(a - 1)}^{2}}{2}) W e k n o w t h e G d i v i d e s O S i n t h e r a t i o 2 : 1 i n t e r n a l l y w h e r e O i s t h e o r t h o c e n t r e o f t h e t r i a n g l e S u p p o s e t h a t O (x, y) (\frac{{(a + 1)}^{2}}{2}, \frac{{(a - 1)}^{2}}{2}) = (\frac{x}{3}, \frac{y}{3}) T h e r e f o r e, t h e o r t h o c e n t e r i s (\frac{3 {(a + 1)}^{2}}{2}, \frac{3 {(a - 1)}^{2}}{2})$

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