Year 12s: what will be the first way you research your future options? >>

parameters

x = a cos^3 (t) , y= a sin^3 (t)
where a is a positive constant 0 < t < 1/2pi

dy/dx= -tan (t)

show that that the equation of the tangent to the curve at the point with parameter t is x sin (t) + y cos (t) = a sin(t) cos (t)

what i tried but in vain,

y = a sin^2(t) sin (t)
x= a cos^2(t) cos (t)

(y - a sin^2(t) sin (t) ) / (x - a cos^2(t) cos (t) ) = - tan (t)
y - a sin^2(t) / x - a cos^2(t) = -1
y - a sin^2(t) = -x + a cos^2(t)
y+x = a(sin^2(t) + cos^2(t) => y+x= a

This is what i got but it is far from the answer required... please need help

also 2nd part of question ask
hence show that , if this tangent meets the x-axis at X and the y-axis at Y , then the length of XY is always equal to a.

(edited 5 years ago)

Reply 1

ghostwalker

Original post by Carlos Nim

(y - a sin^2(t) sin (t) ) / (x - a cos^2(t) cos (t) ) = - tan (t)
y - a sin^2(t) / x - a cos^2(t) = -1

The second quoted line doesn't follow from the first.

Have a think.

Reply 2

Carlos Nim

Original post by ghostwalker

The second quoted line doesn't follow from the first.

Have a think.

ok i got it, but how do i do part 3 now?

X should be ( unknown , 0)
and Y ( 0, unknown)
what value do i take the unknown?

(edited 5 years ago)

Reply 3

Prasiortle

Original post by Carlos Nim

ok i got it, but how do i do part 3 now?

X should be ( unknown , 0)
and Y ( 0, unknown)
what value do i take the unknown?

You need to set e.g. y=0 in your tangent equation to solve for where it crosses the x-axis (make x the subject of the equation).

Reply 4

Carlos Nim

Original post by Prasiortle

You need to set e.g. y=0 in your tangent equation to solve for where it crosses the x-axis (make x the subject of the equation).

i got X ( aCos(t) , 0)
and Y (0, aSin(t) )
some weird coordinates lol

Reply 5

Prasiortle

Original post by Carlos Nim

i got X ( aCos(t) , 0)
and Y (0, aSin(t) )
some weird coordinates lol

Those are correct. Now you just need to find the length of XY using Pythagoras. (To simplify the expression, you'll also need the identity sin^2x + cos^2x = 1, which you should know from C2).