Help with a differentiation problem

Question

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

b) Show that there is no line through the point (2,7) that is tangent to the parabola. Then draw a diagram to see why. 

Answer 1

y = x² + x has derivative y' = 2x + 1. This gives m, the slope of tangent to curve at T(x,y), or T(x,x²+x)...point(s) of tangency.

 

If said tangent lines run through (2,-3), which is a point NOT ON the graph of y, then we can write the line's equation using Point Slope form:

 

y -(-3) = m(x-2); now substitute for y and m:

 

x²+x +3 = (2x+1)(x-2); this simplifies to x² -4x - 5 = 0.  Solutions (x coordinate of point(s) of tangency) are

T₁(5, 30)   T₂(1, 0).

 

You are now able to write the equation of tangents at these two points on the curve.

 

 

As for part b, use the principles employed above but with (2,7) as the "point NOT ON the graph of y"--presumably you can't solve the equation in that case.