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For each real number a, the line y = (2a + 3) x - a^2 intersects the curve y = x^2 + 3x at precisely one point.  Find the line intersecting the curve at the point (2/3, 22/9).

 May 20, 2020
 #1
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We only need to solve this

 

22/9  = (2a + 3)(2/3)  - a^2

 

22/9  =  (4/3)a + 2  - a^2     rearrange as

 

a^2  - (4/3)a  + 22/9 - 2  =  0      

 

a^2  - (4/3)a  + 4/9  =  0      factor as

 

(a - 2/3)^2  =  0

 

a  = 2/3

 

The line  is

 

y = (13/3)x - 4/9

 

See the graph here :  https://www.desmos.com/calculator/oxhhombjut

 

 

cool cool cool

 May 20, 2020

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