Find the value of the constant c for which the line y = 2x + c is a tangent to the curve y2 = 4x.?

1 Answer
Somebody N.
Dec 29, 2017

See below.

Explanation:

In order to find out if a line is tangent to a curve, we can equate the line a curve together. This will give us a quadratic equation in this case. We can then use the discriminant of a quadratic to determine if a line is, tangent, misses the curve completely, or intersects the curve at 2 points.

The discriminant is:

#sqrt(b^2-4ac)#

If:

#sqrt(b^2-4ac)>0# ( the line intersects the curve at 2 points )

#sqrt(b^2-4ac)<0# ( the line misses the curve completely )

#sqrt(b^2-4ac)=0# ( the line touches the curve at 1 point i.e tangent )

From equations:

#y^2=4x#

#y=2x+c#

#:.#

#(2x+c)^2=4x#

#4x^2+4xc +c^2=4x#

#4x^2+4xc-4x +c^2=0#

#4x^2+(4c-4)x +c^2=0#

#sqrt((4c-4)^2-(4(4)(c^2))) =0#

Squaring both sides:

#(4c-4)^2-4(4)(c^2)=0#

#16c^2-32c+16-16c^2=0#

#-32c+16=0=>c=1/2#

Line will be:

#y=2x+1/2#

GRAPH:

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