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Linear Programming Problems

Find the equa...

Question

Find the equation of the circle passing through the points of intersection of the line $x + 3 y = 0$ and $2 x - 7 y = 0$ and whose centre is the point of intersection of the lines $x + y + 1 = 0$ and $x - 2 y + 4 = 0$ .

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Solution

$\begin{matrix} T h e p o i n t o f i n t e r s e c t i o n o f t h e l i n e x + 3 y = 0 a n d 2 x - 7 y = 0 i s (0, 0) L e t (h, k) b e t h e c e n t r e o f c i r c l e w i t h r a d i u s a . t h u s, i t s e q u a t i o n w i l l b e {(x - h)}^{2} + {(y - k)}^{2} = a^{2} T h e p o i n t o f t h e i n t e r s e c t i o n o f t h e l i n e x + y + 1 = 0 a n d x - 2 y + 4 = 0 i s (- 2, 1) ∴ h = - 2, k = 1 ∴ E q u a t i o n o f t h e r e q u i r e d c i r c l e = {(x + 2)}^{2} + {(y - 1)}^{2} = a^{2} - - - - (1) a l s o, e q u a t i o n (1) p a s s e s t h r o u g h (0, 0) ∴ {(0 + 2)}^{2} + {(0 - 1)}^{2} = a^{2} 4 + 1 = a^{2} ∴ a = \sqrt{5} (∴ a > 0) s u b s t i t u t i n g t h e v a l u e s f o a i n e q u a t i o n (1) {(x + 2)}^{2} + {(y - 1)}^{2} = 5 x^{2} + 4 + 4 x + y^{2} + 1 - 2 y = 5 x^{2} + 4 x + y^{2} - 2 y = 0 H e n c e, I t i s t h e r e q u i r e d a n s w e r . \end{matrix}$

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