Parametric (1 Viewer)

[epiktmt14]

Hi all. I was just doing some parametrics question and came across this question that I can't do. Please help

From a point P on the parabola x^2=2y, a tangent is drawn. From the focus S, a perpendicular is drawn to meet the tangent at R
(a) find the equation of SR (I've already got this one)
(b) find the locus of R (need help on this one)

thanks guys

 

[Trebla]

Hi all. I was just doing some parametrics question and came across this question that I can't do. Please help

From a point P on the parabola x^2=2y, a tangent is drawn. From the focus S, a perpendicular is drawn to meet the tangent at R
(a) find the equation of SR (I've already got this one)
(b) find the locus of R (need help on this one)

thanks guys

Check my working...not sure if it is correct...

Given P(p, 0.5p²) and S(0, 0.5)
Equation of perpendicular SR to tangent:
y = - x/p + 0.5
Equation of tangent:
y = px - 0.5p²

Solve simultaneously to find R:
- x/p + 0.5 = px - 0.5p²
x(p + 1/p) = 0.5(1 + p²)
x(1 + p²)/p = 0.5(1 + p²)
x = 0.5p
=> y = 0
Hence we have R(0.5p, 0)
This implies that the locus is the entire x-axis except x = 0 (since p is non-zero for equation of SR to be valid). However, if R is (0, 0), then the tangent would be y = 0 and SR would be x = 0 which is valid hence x = 0 is included in the locus of R making it the entire x-axis.

 

[ninetypercent]

equation of tangent is (1)
equation of SR is (2)

solving simultaneously. substitute (1) in (2) for y



therefore, the locus of R is y = 0, i.e. the x axis

edit: ahhh got beaten

 

[epiktmt14]

hhaha (Y) thanks guys