Answered: find the key values for a conic a. y^2…

Holt Mcdougal Larson Pre-algebra: Student Edition 2012

1st Edition

ISBN:9780547587776

Author:HOLT MCDOUGAL

Publisher:HOLT MCDOUGAL

Chapter9: Real Numbers And Right Triangles

Section9.5: The Distance And Midpoint Formulas

Problem 3C

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find the key values for a conic

a. y^2 - 8x - 4y + 4 = 0

b. 6x^2 - 4y^2 -12x-8y-22=0

c. x^2 +8y^2 +2x-80y+192=0

Expert Solution

Step 1

$(a) y^{2} - 8 x - 4 y + 4 = 0 (y^{2} - 4 y + 4) - 8 x = 0 {(y - 2)}^{2} = 8 x w h e r e y = y - 2, a = 2 w h i c h i s t h e g e n e r a l f o r m o f a p a r a b o l a v e r t e x x = 0, y = 0 \Rightarrow x = 0, y = 2 f o c u s x = 0 + 2, y = 2 \Rightarrow x = 2, x = 2 t h e g i v e n e q u a t i o n i s p a r a b o l a w i t h v e r t e x a t (0, 2) a n d f o c u s a t (2, 2) (b) 6 x^{2} - 4 y^{2} - 12 x - 8 y - 22 = 0 6 (x^{2} - 2 x) - 4 (y^{2} + 2 y) - 22 = 0 6 (x^{2} - 2 x + 1) - 4 (y^{2} + 2 y + 1) - 22 - 6 + 4 = 0 6 {(x - 1)}^{2} - 4 {(y + 1)}^{2} - 24 \frac{{(x - 1)}^{2}}{4} - \frac{{(y + 1)}^{2}}{6} = 1 w e k n o w, \frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1 i s t h e h y p e r b o l a e q u a t i o n w i t h h o w i z o n t a l t r a n s v e r s e a x i s . g i v e n e q u a t i o n i s h y p e r b o l a w h e r e h = 1, k = - 1, a = 2, b = \sqrt{6}$

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