Question

find the equation of the parabola whose axis is parallel to x-axis and which passes through the points (-2,1),(1,2)and (-1,3)

Answer 1

SOLUTION :

TO DETERMINE

The equation of the parabola whose axis is parallel to x-axis and which passes through the points (-2,1),(1,2)and (-1,3)

EVALUATION

Let the equation of the parabola whose axis is parallel to x-axis is

$\sf{ x = a {y}^{2} + by + c\: } \: \: \: ...(1)$

Now Equation (1) passes through the point (-2,1)

So we have

$\sf{ a + b + c = - 2 \: \: \: \: \: ......(2)\: }$

Again Equation (1) passes through the point (1,2)

So we have

$\sf{4 a + 2b + c = 1 \: \: \: \: \: ......(3)\: }$

Again Equation (1) passes through the point (-1,3)

So we have

$\sf{9 a + 3b + c = - 1 \: \: \: \: \: ......(4)\: }$

Equation (3) - Equation (2) gives

$\sf{3a + b = 3 \: } \: \: .....(5)$

Equation (4) - Equation (3) gives

$\sf{ 5a + b = - 2}\: \: \: ......(6)$

Equation (6) - Equation (5) gives

$\sf{ - 2a = 5}$

$\implies \displaystyle \sf{a = - \frac{5}{2} \: }$

From Equation (5)

$\displaystyle \sf{b = 3 + \frac{15}{2} = \frac{21}{2} \: }$

From Equation (2)

$\displaystyle \sf{c = - 2 + \frac{5}{2} - \frac{21}{2} = - 10\: }$

Putting the values of a, b, c in Equation (1) we get the equation of the parabola as

$\displaystyle \sf{x = - \frac{5}{2} {y}^{2} + \frac{21}{2} y - 10\: }$

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