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?

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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?

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Answer 1 · 2018-07-17T13:52:17

$m=2$

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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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

Explanation:

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