There are two quadratic functions and a linear function and contact with each other at only one point, and contact with each other at only one point,
They satisfy the below conditions:
Let be the -coordinate of the point of intersection of and whose -coordinate is in between and
Find the value of
This problem is a part of <Grade 10 CSAT Mock test> series .
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f ( x ) − h ( x ) = 0 has an equal root of α , and since the leading coefficient of f ( x ) is 1 ,
f ( x ) = ( x − α ) 2 + h ( x ) .
Using similar method, we get g ( x ) = 4 ( x − β ) 2 + h ( x ) .
Know that β = 2 α .
Then let's solve f ( t ) = g ( t ) .
( t − α ) 2 + h ( t ) = 4 ( t − β ) 2 + h ( t )
3 t 2 − 1 4 α t + 1 5 α 2 = 0
( 3 t − 5 α ) ( t − 3 α ) = 0 .
Since α < t < 2 α , we get t = 3 5 α .
∴ α 2 1 0 t = 3 5 0 .