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Q. Find the equations of the chords of the parabola $y^{2} = 4 a x$ which pass through the point (- 6a, 0) and which subtends an angle of 45 $°$ at the vertex.

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Solution

Dear student,

$T h e v e r t e x o f p a r a b o l a y^{2} = 4 a x i s (0, 0) E q u a t i o n o f a c h o r d i n t h e s l o p e - i n t e r c e p t f o r m i s y = m x + c . . . (1) h e r e, m = \tan 45 ° = 1 (a s t h e c h o r d s u b t e n d s a n a n g l e 45 ° a t t h e v e r t e x) T h u s e q u a t i o n (1) b e c o m e s y = 1 . x + c \Rightarrow y = x + c . . . (2) O n t h e o t h e r h a n d, t h i s l i n e p a s s e s t h r o u g h t h e p o i n t (- 6 a, 0), c o n s e q u e n t l y i t s c o o r d i n a t e s s a t i s f y e q u a t i o n (2) : 0 = - 6 a + c \Rightarrow c = 6 a t h u s, y = x + 6 a i s t h e r e q u i r e d e q u a t i o n o f t h e c h o r d .$
Regards

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