maths question

hi, please could i have help on part c
The ms says we need to find a lower limit, not sure what that means:
question: https://ibb.co/KcfCgDtn
ms: https://ibb.co/Hp93qNdW
thanks!

If you're asking about the minimum value specifically, imagine sliding that normal line down vertically - there are points where the line crosses the curve at 2 points and then it will reach a point where it intersects once. Since you're equating a quadratic to a line, this is the usual "discriminant = 0" problem.

But when the discriminant=0, isn’t there only one root and the questions asks for two distinct roots so shouldn’t b2-4ac>0? This is shown in the upper limit but I don’t get the lower limit part. Thank you.

Youve found the value of k (y intercept) where the line touches the curve. So when k > that value ....
Youd get the same result solving b^2-4ac>0 but its usually less hassle to solve an equation rather than an inequality and interpret it appropriately.

Sorry, I’m still quite confused. I’ve attached my working below through solving but I don’t get how you can get two k values because there is no quadratic?

just getting your pencil out and sliding it up and down the y axis (different values of k) and with a gradient of -1/2 should convince you that there is only one value of k where the pencil touches the curve and it corresponds to the lower value of k. So your working is correct in that k>43/8 corrresponds to the range of k where the l intersects the quadratic. So Im not sure what youre confused about?
You stated in the OP you were only asking about the lower value, but the upper value for k (two values?) is when l intersects with the starting point of the parametric curve, as should be clear from the given diagram and moving your pencil up and down the y-axis with a grad of -1/2. The curve C is a quadratic function but only defined on a restricted domain as should be clear from the earlier parts.

Wait so my working is correct and you don’t need the lower limit?

Im a bit confused about what youre asking but k>43/8 is correct (see the ms). You can find it as an equality discriminant=0, to get 43/8 then put the > in. Or solve the discriminant>0 as youve done. Personally I think the ms way is a bit simpler/less error prone but theres not much in it.

Sorry if I’m being unclear, basically the red part I didn’t do but I don’t get why they are doing that, the green part is fine

I realize its that part, but you get the same ans now k>43/8 from disciminant>0 whereas the ms does discriminant=0 and then replaces the = with a >. So what part is unclear?
Loosely speaking theyve slid their pencil down the y axis from the upper bound where the two intersections start and noted that the two solutions become one when the pencil is tangent to the curve (discriminant equals zero). As far as I can see it, this is the same logic that youve done.

Oh I see, but why did they get 13/2? I only got 43/8 as you can see in my working but it’s only one solution?