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
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 $\displaystyle a$