find equations for two lines through one point, tangent to a parabola?

Question

find equations for both lines through the point (2, -27), that are tangent to the parabola y=x^2 +9x

Answer 1

y = x² + 9x

y' = 2x + 9

y + 27 = 2x(x - 2)

y + 27 = 2x² - 4x

y = 2x² - 4x - 27 (and y = x² + 9x)

2x² - 4x - 27 = x² + 9x

x² - 13x - 27 = 0

x = 14.8 and -1.8

So, y = 352 and -13 (I rounded these values)

y - 352 = 29.6(x - 14.8)

y + 13 = -3.6(x + 1.8)


Here are your equations:

y = 29.6x - 86.1

y = -3.6x - 19.5

Answer 2

The equation of a line having slope m and passing by utilising ability of way of (2, -3) is y + 3 = m(x - 2) ... ( a million ) fixing it with the equation of the parabola, m(x - 2) - 3 = x^2 + x => x^2 + (a million - m)x + 2m + 3 = 0 For the line to be the tangent to the parabola, the above equation in x will could desire to have the comparable 2 roots => its discriminant = 0 => (a million - m)^2 - 8m - 12 = 0 => m^2 - 10m - 11 = 0 => (m -11) (m + a million) = 0 => m = 11 or - a million Plugging m = 11 and m = -a million in eqn. ( a million ) provides you the two equations of the tangents as y + 3 = 11(x - 2) and y + 3 = - a million * (x - 2) => y = 11x - 25 and y = - x - a million For the given parabola, y - x^2 - x = 0, no tangent must be drawn from a ingredient (x1, y1) that's indoors the concave area of the parabola and the condition to make optimistic that's y1 - x1^2 - x1 > 0 For the ingredient (x1, y1) = (2, 7), the is 7 - 2^2 - 2 = a million > 0 and is optimistic utilising this that no tangent is in lots of cases drawn from that ingredient to the parabola.