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Given four points P1,P2,P3a n dP4 on the...

Given four points `P_1,P_2,P_3a n dP_4` on the coordinate plane with origin `O` which satisfy the condition `( vec O P)_(n-1)( vec+O P)_(n-1)=3/2 vec O P_(n)` . i. If `i i . iii. iv. P_v .1v i . vii. a n dv i i i . P_i x .2x . xi. xii.` xiii. lie on the curve `x i v . xv. x y=1, xvi.` xvii. then prove that `x v i i i . xix. xx. P_x x i .3x x i i . xxiii. xxiv.` xxv. does not lie on the curve. xxvi. If `x x v i i . xxviii. xxix. P_x x x .1x x x i . xxxii. ,x x x i i i . P_x x x i v .2x x x v . xxxvi. a n dx x x v i i . P_x x x v i i i .3x x x i x . xl.x l i .` xlii. lie on the circle `x l i i i . xliv. xlv. x^(x l v i .2x l v i i .)x l v i i i .+x l i x . y^(l .2l i .)l i i .=1, liii.` liv. then prove that `l v . lvi. lvii. P_l v i i i .4l i x . lx. lxi.` lxii. also lies on this circle.

Text Solution

Verified by Experts

(i) Put n=2 in `vec(OP)_(n-1)+ vec(OP) _(n+1)= (3)/(2) vec(OP)_n`
`" "vec(OP)_3= (3)/(2) vec(OP)_2- vec(OP)_1" "`(i)
`" "vec(OP)_1= ahati+ (1)`
`{:(vec(OP)_1= ahati+ (1)/(a) hatj),(vec(OP)_2= bhati+ (1)/(b)hatj):}}ab ne 0`
`therefore " "vec(OP)_3 = (3)/(2) (bhati+ (1)/(b) hatj)- (ahati+ (1)/(a) hatj)`
`" "= ((3b)/(2)-a) hati+ ((3)/(2b) - (1)/(a) )hatj`
If `P_3` lies on xy=1, we have
`" "((3b)/(2)-a)((3)/(2b)- (1)/(a))= 1`
or `" "(3b-2a)(3a-2b) = 4ab`
or `" "9ab-6b^(2) - 6a^(2) + 4ab = 4ab`
or `" "2a^(2) - 3ab+ 2b^(2)=0`
which is not possible as Discriminant `lt 0 (a= 0 and b=0` not possible)
(ii) `vec(OP) = cos alpha hati+ sinbeta hatj`
`" "vec(OP)_2 = cos beta hati+ sinbetahatj`
`therefore " "vec(OP)_3= (3)/(2) (cosbeta hati+ sinbeta hatj) - (cosalpha hati+ sin alpha hatj)`
`" "= ((3)/(2) cos beta - cos alpha)hati + ((3)/(2) sin beta-sinalpha) hatj`
Since `P_3` lies on `x^(2) + y^(2)=1`, we have
`((3)/(2) cos beta - cosalpha)^(2) + ((3)/(2) sinbeta-sinalpha)^(2)=1`
or `" "(9)/(4) + 1-3(cosbeta cosalpha + sin beta sinalpha )=1`
or `" "(9)/(4) -3 cos (beta-alpha)=0 rArr cos (beta -alpha)=(3)/(4)" "` (ii)
Put n=3 in the given relation.
`vec(OP)_2 + vec(OP)_4 = (3)/(2) vec(OP)_3`
or `" " vec(OP)_4 = (3)/(2) vec(OP)_3-vec(OP)_2`
`" "=(3)/(2) ((3)/(2) vec(OP)_2-vec(OP)_1)-vec(OP)_2`
`" "=(5)/(4) vec(OP)_2- (3)/(2) vec(OP)_1`
`" "= (5)/(4) (cosbeta hati+sin betahatj) - (3)/(2) (cosalpha hati+ sin alpha hatj)`,
Which lies on `x^(2)+y^(2)=1`
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