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

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 mis 2