Find the equations to both line

Question

Find the equations to both lines through the point (2,1) that are tangent to the parabola y=x²+x+4

y= (smaller slope)

y=(larger slope)

Answer 1

y=x2+x+4

first find the derivative of the function:

dy/dx = 2x+1

The slope of the tangent at point (2,1)  is 5

then use the form:

(Y-Y1)=m(X-X1)

(Y-1) = 5 (X-2), which gives:

Y - 5x +9 = 0

 

Answer 2

First find the derivative of y.

derivative of y is 2x+1

and notice that derivative is slope of y.

Then find two points on the two tangent lines to y using:

(x,x2+x+4) and (2,1). This is to show (x,f(x)) is a point on the original equation f(x). Find the slope of these two points and equate it to the derivative (slope) of the original equation.

therefore 2x+1= (x2+x+4)-1 / x-2 to get x=5 or x=-1.

 

at x=5 the slope is 2(5)+1=11 and the point is (5,34) and the equation of the first tangent line is y-34=11(x-5)

 

at x=-1, the slope is 2(-1)+1=-1 and the point is (-1,4) and the equation of the second tangent line is 

y-4= -1(x+1). It shows that every quadratic equation has two tangent lines from a point that is not on the graph of the quadratic equation.