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a-Normal-to-any-point-on-the-hyperbola-XY-C-meet-the-x-axis-at-A-and-tangents-meets-the-y-axis-at-B-find-the-locus-of-the-mid-point-of-AB-b-find-the-equation-of-assymptotes-of-i-x-2-4-y-2-5-1




Question Number 50979 by peter frank last updated on 23/Dec/18
a)Normal to any point on  the hyperbola XY=C  meet the x−axis at A  and tangents meets  the y−axis at B.find the  locus of the mid point of AB  b)find  the equation of   assymptotes of  (i)(x^2 /4)−(y^2 /5)=1  (ii)(((x−1)^2 )/(16))−(((y−3)^2 )/9)=1
a)NormaltoanypointonthehyperbolaXY=CmeetthexaxisatAandtangentsmeetstheyaxisatB.findthelocusofthemidpointofABb)findtheequationofassymptotesof(i)x24y25=1(ii)(x1)216(y3)29=1
Commented by ggururajguru0219@gmail.com last updated on 23/Dec/18
vg5n
vg5n
Answered by mr W last updated on 23/Dec/18
(a)  xy=c  y′=−(c/x^2 )  P(h,(c/h))  y′=−(c/h^2 )  tangent:  y=(c/h)−(c/h^2 )(x−h)  B(0,y_B )  ⇒y_B =((2c)/h)  normal:  y=(c/h)+(h^2 /c)(x−h)  A(x_A ,0)  0=(c/h)+(h^2 /c)(x_A −h)  ⇒x_A =h−(c^2 /h^3 )  mid point of AB: (p,q)  q=(y_B /2)=(1/2)×((2c)/h)=(c/h)  ⇒h=(c/q)  p=(x_A /2)=(1/2)(h−(c^2 /h^3 ))=(1/2)((c/q)−(q^3 /c))  2cpq=c^2 −q^4   ⇒2cxy=c^2 −y^4
(a)xy=cy=cx2P(h,ch)y=ch2tangent:y=chch2(xh)B(0,yB)yB=2chnormal:y=ch+h2c(xh)A(xA,0)0=ch+h2c(xAh)xA=hc2h3midpointofAB:(p,q)q=yB2=12×2ch=chh=cqp=xA2=12(hc2h3)=12(cqq3c)2cpq=c2q42cxy=c2y4
Commented by mr W last updated on 23/Dec/18
Commented by peter frank last updated on 23/Dec/18
thank you
thankyou
Commented by peter frank last updated on 23/Dec/18
sir please check third line  P(h,(c/h)) it should  be P(ch,(c/h))
sirpleasecheckthirdlineP(h,ch)itshouldbeP(ch,ch)
Commented by mr W last updated on 23/Dec/18
what is your curve?  xy=c or xy=c^2  ?    if xy=c as the question says, then  the point is P(h,(c/h)) because h×(c/h)=c.
whatisyourcurve?xy=corxy=c2?ifxy=casthequestionsays,thenthepointisP(h,ch)becauseh×ch=c.
Commented by peter frank last updated on 23/Dec/18
very sorry sir  found my mistake.my  xy=c^2  insteady of xy=c
verysorrysirfoundmymistake.myxy=c2insteadyofxy=c
Answered by peter frank last updated on 23/Dec/18
b)  (x^2 /4)−(y^2 /5)=1  a=2   b=(√5)  y=±(b/a)x  y=±((√5)/2)  ii) (((x−1)^2 )/(16))−(((y−3)^2 )/9)=1  a=4    b=3  p=1   q=3  y−q=±(b/a)(x−p)  y−3=±(3/4)(x−1)  y=(3/4)x−(9/4) or −(3/4)x+(9/4)
b)x24y25=1a=2b=5y=±baxy=±52ii)(x1)216(y3)29=1a=4b=3p=1q=3yq=±ba(xp)y3=±34(x1)y=34x94or34x+94
Answered by peter frank last updated on 23/Dec/18
a) equation of normal  at xy=c  y−((xt^2 )/c)+(t^3 /c)−(c/t)=0  A(x,0)⇒A(t−(c^2 /t^3 ),0)  from tangent at xy=c  yt^2 +cx−2ct=0  B(0,y)⇒(0,((2c)/t))  A(t−(c^2 /t^3 ),0),B(0,((2c)/(t  )))  x,y)=(((x_1 +x_2 )/2),((y_1 +y_2 )/2))  (x,y)=(((ct)/2)−(c/(2t^3 ))),(c/y)  x=(c/2)−(c/(2t^3 ))......(i)  t=(c/y).......(ii)  sub ii  in i  2x=(c/y)−(y^3 /c)  y^4 +2xcy−c^2 =0
a)equationofnormalatxy=cyxt2c+t3cct=0A(x,0)A(tc2t3,0)fromtangentatxy=cyt2+cx2ct=0B(0,y)(0,2ct)A(tc2t3,0),B(0,2ct)x,y)=(x1+x22,y1+y22)(x,y)=(ct2c2t3),cyx=c2c2t3(i)t=cy.(ii)subiiini2x=cyy3cy4+2xcyc2=0
Commented by mr W last updated on 23/Dec/18
please check eqn. of normal:  y−xt^2 +ct^3 −(c/t)=0  it should be, i think,  y−((xt^2 )/c)+(t^3 /c)−(c/t)=0
pleasecheckeqn.ofnormal:yxt2+ct3ct=0itshouldbe,ithink,yxt2c+t3cct=0
Commented by peter frank last updated on 23/Dec/18
 Mrw please check right or wrong
Mrwpleasecheckrightorwrong
Commented by mr W last updated on 23/Dec/18
please check eqn. of tangent:  yt^2 +x−2ct=0pp  it should be, i think,  yt^2 +cx−2ct=0
pleasecheckeqn.oftangent:yt2+x2ct=0ppitshouldbe,ithink,yt2+cx2ct=0
Commented by mr W last updated on 23/Dec/18
please check:  the final eqn. should be, i think,  y^4 +2cxy−c^2 =0
pleasecheck:thefinaleqn.shouldbe,ithink,y4+2cxyc2=0
Commented by peter frank last updated on 23/Dec/18
equation of tangent and  normql  to xy=c at (ct,(c/t))  x=ct  y=(c/t)  y^′ =−(1/t^2 )  ((y−(c/t))/(x−ct))=−(1/t^2 )  yt^2 +x−2ct=0  normal to xy=c  y^′ =t^2   ((y−(c/t))/(x−ct))=t^2   xt^2 −ct^3 =y−(c/t)  xt^2 −ct^3 +(c/t)=y
equationoftangentandnormqltoxy=cat(ct,ct)x=cty=cty=1t2yctxct=1t2yt2+x2ct=0normaltoxy=cy=t2yctxct=t2xt2ct3=yctxt2ct3+ct=y
Commented by peter frank last updated on 23/Dec/18
Mrw sir please check
Mrwsirpleasecheck
Commented by mr W last updated on 23/Dec/18
if xy=c, then  at point (t,(c/t)), not at (ct,(c/t))  y′=−(c/x^2 )=−(c/t^2 )  ......
ifxy=c,thenatpoint(t,ct),notat(ct,ct)y=cx2=ct2
Commented by peter frank last updated on 23/Dec/18
your absolute right
yourabsoluteright

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