Problem on mid point formula

[maths_hero]

This is the question and solution from the quiz.
I approached the sum in a bit different way:
I derived the tangent formula as y=x-3 as above and then substituted it in the curve's equation.
I got: λ²x² + x(-5λ+1) +1=0
We know point R is mid point of the point of intersections of the line and curve so, 5λ-1=4 => λ=1
And λ=1/4 doesn't appear here. Why is the extra solution of λ=1/4 appearing while using the mid-point formula?

 

[Dr.Peterson]

Because their answer is wrong, for one thing! Have you tried graphing the parabola for [MATH]\lambda = \frac{1}{4}[/MATH]? The line does not intersect the parabola, so there is no chord!

I can't follow their work; what do they mean by "[MATH]T = S_1[/MATH]"? Please tell us what you were taught about these.

Whatever they are doing evidently introduced an extraneous solution, and they failed to check it!

 

[maths_hero]

I can't follow their work; what do they mean by "[MATH]T = S_1[/MATH]"? Please tell us what you were taught about these.

Yes, now I too tried putting λ=1/4 into the curve equation and verified it doesn't work. T=S1 is for chord with mid point formula; here T is the tangent form of a equation and S1 is putting in the values of a point into the equation. SS1=T², T=0 are other equations which we were taught. But again I've no idea when these formulas will fail like what happened here.

 

[Dr.Peterson]

I'm not familiar with either of these formulas; if you could show me how they were taught, or give a reference online, I could judge better.

But it is not uncommon for an equation to be true if some condition holds, but not "only if" -- that is, it can be true in other cases as well. That is ultimately the cause of any extraneous solution. I suspect this one may involve complex-number "intersections", which are not visible on the graph, and so are not relevant to the problem.