Question

Show that the line y=-3x+2 is tangent to the parabola y^2=-24x, and find the point of contact

Answer 1

@mathmate

Answer 2

@ganeshie8

Answer 3

hints:
1. calculate the derivative of the parabola (using implicit differentiation).
The result could be in terms of x, or y, or both, call it y'.
2. Calculate the point of intersection of the line and the parabola.
you can do that by substitution.
y^2=-24x..........(1)
y=-3x+2...........(2)
You can substitute y from (2) into (1), but with some algebraic effort, or
solve for x in (2) and substitute the value of x (in terms of y) into (1), which will take a little less algebraic work.
If you find a double root, it is sign that the line is tangent to the parabola.
Back substitute x or y into equation (2) to find the other variable, thus obtain the point of tangency (contact) as (xt, yt).
3. Substitute (xt,yt) into the expression of y' in step 1 and compare with the slope of the line. If they are the same, the line is tangential to the parabola.

Answer 4

2yy'=-24
y'=-24/2y
y'=-12/y

Answer 5

y'=-12/sq.rt -24x

Answer 6

You don't need to solve in terms of x, because you will know the value of y after solving the next step.

Answer 7

(-3x+2)^2=-24x
9x^2-12x+4=-24x
9x^2+12x+4=0
9(x+2/3)^2-4+4=0
9(x+2/3)^2=0
(x+2/3)^2=0
x+2/3=0
x=-2/3

Answer 8

y^2=-24x
y^2=16
y=4 or y =-4

Answer 9

y'=-12/y
y'=-12/4
y'=-3

and y'=-12/-4
y'=3

Answer 10

is that correct so far?

Answer 11

r u there?

Answer 12

@mathmate

Answer 13

I'll take a look, was on the phone, sorry.

Answer 14

ok

Answer 15

Yep, so far so good.
But you're on thin ice from here on! :)

Answer 16

Would you like to continue?

Answer 17

yes but im not sure what i just find.. im thinking its the point of intersection .. where the line and the parabola function meet ..am i right?

Answer 18

yes, but remember, the curves look like this:

So you have to choose your valid solution carefully.

Answer 19

would u say that the point y=-4 and y=4 are double root? or would it be y'=3 and y'=3 the double root ?