Find the equation of the parabola: vertex at (3,-1), axis parallel to the y- axis and the line 4x + y - 7= 0 is a tangent

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- Mar 20th 2010, 04:14 AMdeltaxrayparabola and tangent
Find the equation of the parabola: vertex at (3,-1), axis parallel to the y- axis and the line 4x + y - 7= 0 is a tangent

Thanks - Mar 20th 2010, 05:46 AMskeeter

given the vertex and direction of the axis of symmetry, the equation of the parabola is ...

$\displaystyle y = a(x-3)^2 - 1$ , where $\displaystyle a$ is a constant TBD.

since the line $\displaystyle y = -4x+7$ is tangent to the parabola ...

(1) the line and parabola are equal at the point of tangency

$\displaystyle a(x-3)^2 - 1 = -4x+7$

(2) the slope of the parabola at the point of tangency equals the slope of the line

$\displaystyle 2a(x-3) = -4$

solve the system of equations for $\displaystyle x$ and $\displaystyle a$