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Question 31479: I'm stuck with this problem, please help me?
Take points A(7/2,0), B(0,7) and C(-7/6,0) on the xy-plane. The parabola y=-x^2+ax+b is tagent to both lines BA and BC.
1) Determine a and b
2) Calculate the area of the domain bounded by the line BA, the parabola and the y-axis.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Take points A(7/2,0), B(0,7) and C(-7/6,0) on the xy-plane. The parabola y=-x^2+ax+b is tagent to both lines BA and BC.
1) Determine a and b
2) Calculate the area of the domain bounded by the line BA, the parabola and the y-axis.
SLOPE OF AB = (7-0)/(0-7/2)= -2
SLOPE OF BC = (7-0)/(0+7/6)=6
BA AND BC ARE TANGENTS THROUGH A COMMON POINT B.HENCE EQN.OF CHORD OF CONTACT OF B(0,7) IS
(Y+7)/2= -X*0+A(X+0)/2+B
Y+7=AX+2B
OR
..Y=AX+2B-7
.I
THIS MEETS PARBOLA
.Y= -X^2+AX+B
... AT
AX+2B-7= -X^2+AX+B
X^2=7-B
.OR
X=+(7-B)^0.5 AND -(7-B)^0.5
FOR
X=(7-B)^0.5
..Y=A(7-B)^0.5+2B-7
.LET THIS BE POINT P ON THE CURVE.
FOR
.X= -(7-B)^0.5
Y= -A(7-B)^0.5+2B-7
..LET THIS BE POINT Q ON THE CURVE.
SLOPE OF BAP = -2 ={A(7-B)^0.5+2B-7-7}/{(7-B)^0.5-0}
-2(7-B)^0.5=A(7-B)^0.5+2B-14
..OR
(A+2)(7-B)^0.5+2B-14=0
..II
SLOPE OF BCQ =6={-A(7-B)^0.5+2B-7-7}/{-(7-B)^0.5-0}
-6(7-B)0.5=-A(7-B)^0.5+2B-14
..OR
(6-A)(7-B)^0.5+2B-14=0
..III
EQN.II - EQN.III
GIVES
(2A-4)(7-B)^0.5=0
.2A=4
..OR
A=2
SUBSTITUTING IN EQN.II
WE GET
4(7-B)^0.5=2(7-B)
2=(7-B)/(7-B)^0.5=(7-B)^0.5
7-B=4
OR B=3
.HENCE EQN. OF CURVE IS Y= -X^2+2X+3=4-(X-1)^2
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EQN. OF BA IS Y-7= -2(X-0)......Y=7-2X
HOPE YOU KNOW INTEGRATION .THE SECOND BIT NEEDS INTEGRATION.FIRST SEE THE GRAPH
THE QUESTION IS AREA OF DOMAIN BOUNDED BY Y AXIS WHOSE EQN. IS X=0....
BA LINE WHOSE EQN.IS....Y=7-2X......AND PARABOLA WHOSE EQN.IS..Y=4-(X-1)^2..
THIS IS NOT VERY CLEAR...YOU CAN SEE ROM GRAPH...THIS IS OPEN TO DIFERENT INTERPRETATIONS.IF IT IS MADE CLEAR WE CAN DO IT USING INTEGRATION...
FOR EX. THE QUESTION COULD BE AREA BETWEEN Y AXIS,BA LINE AND THE CURVE UPTO TANGENT POINT P AS WE CALLED IT.P IS (2,3)....
BELOW THIS TANGENT POINT WHAT DOES THE DOMAIN BETWEEN BA,Y AXIS AND PARBOLA HAS NO MEANING..EITHER WE CAN HAVE DOMAIN BETWEEN BA AND PARABOLA ...OR...BETWEEN BA AND Y AXIS ...OR BETWEEN Y AXIS AND PARABOLA.SO PLEASE CHEKUP AND COME BACK TO KNOW THE SOLUTION.ALSO CONFIRM YOU KNOW INTEGRATION.
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