# Math Help - parabola and tangent

1. ## parabola 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

2. Originally Posted by deltaxray
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 ...

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

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

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

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

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

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

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