Drawing two perpendicular tangent line from the origin

Javion Kerr

Javion Kerr

Answered question


Drawing two perpendicular tangent line from the origin to y=x22x+a.
We drew two perpendicular tangent line from the origin (the point (0,0) ) to the curve y=x22x+a, what is the value of a?
1) 54
2) 54
3) 34
4) 34

Answer & Explanation



Beginner2022-04-06Added 15 answers

Let the tangent be y=mx, (as it passes through the origin), then we have the quadratic x2(m+2)x+44a=0. In order for the given line be tangent to the given parabola, it must intersect at only one point, and hence B24AC=0). So we get a quadratic for m as m2+2m4(1a)=0. This means that two tangents are possible, and they will be perpendicular if m1m2=14(1a)=1a=54 (product of roots of a quadratic ax2+bx+c=ca)
Ishaan Stout

Ishaan Stout

Beginner2022-04-07Added 14 answers

Let f(x)=x2-2x+a.
A line passing through (0,0) is of the form y=λx. If such a line is tangent to the graph of f at some point (α,β), then:
1. β=f(α)=α22α+a;
2. λ=f(α)=2α2.
Therefore, β=λα=2α22α and β=α22α+a,
and therefore . So, α=a or α=a. In the first case, β=2a2a, and in the second case β=2a+2a. In the first case, the slope of the tangent line will be 2a2 and in the second case it will be 2a2. The only situation in which the tangent lines will be orthogonal, that is, the only case in which the product of these numbers is -1, is when a=±54
Hence, a=54.

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