# Find Equations Of Both Lines Through The Point (2, −3) That Are Tangent To The Parabola Y = X2 + X.

Find the equation of both lines that pass through the point (2, -3) and are tangent to the parabola y = x^2 + x.

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The equation of the line through (2, -3) and tangential to the parabola y = x^2 + x has to be determined.

The slope of a tangent to a curve y = f(x) at the point x = c is f'(c)

y' = 2x + 1

Let the line that passes through (2, -3) be a tangent to the parabola at (x, x^2 + x)

(x^2 + x + 3)/(x - 2) = 2x + 1

=> x^2 + x + 3 = 2x^2 - 4x + x - 2

=>  x^2 - 4x - 5 = 0

=> x^2 - 5x + x - 5 = 0

=> x(x - 5) + 1(x - 5) = 0

=> (x + 1)(x - 5) = 0

x = -1 and x = 5

The tangents are lines passing through (2, -3) and (-1, 0) and through (2, -3) and (5, 30)

The lines are (y + 3)/(x - 2) = (3/-3) = -1

=> y + 3 = 2 - x

=> x + y + 1 = 0

and (y + 3)/(x  - 2) = (33/3)

=> y + 3 = 11x - 22

=> 11x - y - 25 = 0

The equation of the two tangent lines to the curve y = x^2 + x passing through (2, -3) are x + y + 1 = 0 and 11x - y - 25 = 0