What are the coordinates of the point of intersection of the tangents when the equation of the tangent to the curve #y= x^2 -6x + 5# at the points, #x = 1# and #x = 5# is/are found?
1 Answer
The tangent equation are
# y = -4x+4 # and# y = 4x-20 #
Which intersect at
Explanation:
We have:
# y= x^2 -6x + 5#
The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point.
Differentiating wrt
# dy/dx = 2x - 6#
So when
# { (y= 1-6+5, = 0), (y' = 2 - 6, = -4) :} => { (((0,0))), (m=-4) :} #
So, the equation of the first tangent, Using the point/slope form
# y - 0 = -4 (x-1) => y = -4x+4 #
And, when
# { (y= 25-30+5, = 0), (y' = 10-6, = 4) :} => { (((5,0))), (m=4) :} #
So, the equation of the second tangent is;
# y - 0 = 4 (x-5) => y = 4x-20 #
And, the point of intersection is the simultaneous solution so:
# y = -4x+4 # and# y = 4x-20 #
Adding gives:
Substitution gives,
Thus the coordinate of the insertion of the tangents is
We can confirm this graphically:
graph{(y-(x^2 -6x + 5))(y-(-4x+4))(y-(4x-20))=0 [-3, 8, -15, 15]}