A point on the parabola whose axis is paralle | vedclass

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(A) The general equation of a parabola with its axis parallel to the $X$-axis is $x = ay^2 + by + c$.Since the parabola passes through $(0,1)$,we have

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$(3,-1)$

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(A) The general equation of a parabola with its axis parallel to the $X$-axis is $x = ay^2 + by + c$.Since the parabola passes through $(0,1)$,we have $0 = a(1)^2 + b(1) + c$,so $a + b + c = 0$.Since it passes through $(0,-2)$,we have $0 = a(-2)^2 + b(-2) + c$,so $4a - 2b + c = 0$.Subtracting the two equations: $(4a - 2b + c) - (a + b + c) = 0 \implies 3a - 3b = 0 \implies a = b$.Substituting $b = a$ into $a + b + c = 0$,we get $2a + c = 0$,so $c = -2a$.Since the parabola passes through $(3,0)$,we have $3 = a(0)^2 + b(0) + c$,so $c = 3$.Thus,$c = 3$,$2a = -3 \implies a = -\frac{3}{2}$,and $b = -\frac{3}{2}$.The equation is $x = -\frac{3}{2}y^2 - \frac{3}{2}y + 3$,or $2x = -3y^2 - 3y + 6$.Testing the options:For $(A) (3,-1)$: $3 = -\frac{3}{2}(-1)^2 - \frac{3}{2}(-1) + 3 = -\frac{3}{2} + \frac{3}{2} + 3 = 3$. This point lies on the parabola.

$A$ point on the parabola whose axis is parallel to the $X$-axis and which passes through the points $(0,1), (3,0), (0,-2)$ is

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