Question

the curve y=ax^2 +bx+c passes through the points (1,8),(-1,2),(2,14). Find the value of a,b,c and thus the eqn of the line.
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Answer 1

Step-by-step explanation:

The curve y=ax^{2}+bx+cy=ax

2

+bx+c passes through the points (1, 8), (0, 5)\ and\ (3, 20)(1,8),(0,5) and (3,20) .

The co-ordinates given above will satisfy the equation:

y=ax^{2}+bx+c ...... (1)y=ax

2

+bx+c......(1)

First co-ordinate (1,8):

Putting x = 1\ and\ y = 8x=1 and y=8

\begin{lgathered}8 = a \times 1^{2} + b \times 1 + c\\\Rightarrow 8 = a + b + c ...... (2)\end{lgathered}

8=a×1

2

+b×1+c

⇒8=a+b+c......(2)

Second co-ordinate (0,5):

Putting x =0\ and\ y = 5x=0 and y=5

\begin{lgathered}5 = a \times 0^{2} + b \times 0 + c\\\Rightarrow c = 5 ...... (3)\end{lgathered}

5=a×0

2

+b×0+c

⇒c=5......(3)

Putting c = 5c=5 in equation (2):

\Rightarrow a + b = 3 ...... (4)⇒a+b=3......(4)

Third co-ordinate (3, 20):

Putting x =3\ and\ y = 20x=3 and y=20

20 = a \times 3^{2} + b \times 3 + c20=a×3

2

+b×3+c

Putting value c = 5c=5

\begin{lgathered}\Rightarrow 20 = 9a + 3b + 5\\\Rightarrow 3a + b = 5...... (5)\end{lgathered}

⇒20=9a+3b+5

⇒3a+b=5......(5)

Using elimination method in equation (4) and (5)

Subtracting equation (4) from equation (5):

\begin{lgathered}\Rightarrow 2a = 2\\\Rightarrow a = 1\end{lgathered}

⇒2a=2

⇒a=1

From equation (4):

\Rightarrow b = 2⇒b=2

Hence a = 1, b =2\ and\ c = 5a=1,b=2 and c=5

The equation of the curve is:

\Rightarrow y = x^2 + 2x + 5⇒y=x

2

+2x+5