Few past paper questions i cant do (1 Viewer)

[ek102]

1. For what values of 'b' is the line y=3x+b, a tangent to y=X^3 ?
2. AB is a chord of length 16 units of the circle x^2 + y^2= 100 and P is the midpoint of AB. Determine the locus of P.
3. b) of attachement

thanks in advance

 

[dan964]

For Q3, find gradient of tangent of P (easy)
find equation of OP using the point O and the point P
find the point T by intersecting OP with directrix
find gradient of ST, and show that is same as above gradient.

 

[jathu123]

for q1
for two curves to be tangent, their gradients must be equal at their points of contact. We know the gradient of the line is 3, so the gradient of the curve must also be 3 at the points of contact. So find the derivative of x^3 and find the solutions for which it is equal to 3. Now these are the x-coordinates of the point of contact, sub this into y=x^3 to find its y- value. Now finally sub these into the line y=3x+b and solve for b. I have provided the working out if you need it

let f(x)=x^3
f'(x)=3x^2
3x^2=3 --> x = 1 or -1
sub the into f(x), f(1) = 1, f(-1) = -1
∴ the points of contact are (1,1) and (-1,-1). This means that these points must also satisfy the equation of the line.
sub each of this back into the line to find b;
1=3+b or -1=-3+b
b=-2 or 2

 

[dan964]

For 2, isn't r=10?, which makes me think how can you have a chord longer than the radius?

 

[jathu123]

For 2, isn't r=10?, which makes me think how can you have a chord longer than the radius?

I think you got confused with the diameter

 

[dan964]

I think you got confused with the diameterView attachment 33461

which means the answer is yes.
in which case draw a diagram

 

[ek102]

Thanks alot, im still lost for the circle locus question though, i have no idea where to start on that question, do i use a cartesian equation to solve it?

 

[InteGrand]

Thanks alot, im still lost for the circle locus question though, i have no idea where to start on that question, do i use a cartesian equation to solve it?

Note that the midpoint of a chord of fixed length on a given circle will always be a fixed distance from the centre (in this case, that distance is 6, which follows from Pythagoras and recalling that the line joining the circle centre and the chord's midpoint is perpendicular to the chord).

The locus of P is a circle centred at O with radius 6.