There are two values of m where the line y=mx−1 is tangent to the parabola y=(x−2)^2+4. One of the possible values of m is positive, one is negative. How do I find the positive value of m?

1 Answer
Harish Chandra Rajpoot
Jul 17, 2018

#m=2#

Explanation:

Setting #y=mx-1# in the equation of parabola: #y=(x-2)^2+4#, we get

#mx-1=(x-2)^2+4#

#mx-1=x^2-4x+4+4#

#x^2-(m+4)x+9=0#

Since, the given line: #y=mx-1# is tangent to the parabola at a single point i.e. above equation must have equal real roots i.e. the determinant: #B^2-4AC# of above quadratic equation must be zero as follows

#(-(m+4))^2-4(1)(9)=0#

#m^2+8m+16-36=0#

#m^2+8m-20=0#

#m^2+10m-2m-20=0#

#m(m+10)-2(m+10)=0#

#(m+10)(m-2)=0#

#m+10=0\ \ or\ \ m-2=0#

#m=-10\ \ or\ \ m=2#

hence, the positive value of #m#is #2#

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