Flag Analytical Geometry> A, B are fixed points; AP and BQ are para...
question mark

A, B are fixed points; AP and BQ are parallel chords of a variable circle such thatAP/BQ is a constant. Prove that the locus of P is a circle.

Uke , 3 Years ago
Grade 11
anser 1 Answers
Yuvraj Singh

Last Activity: 2 Years ago

Ans:
We takeOas origin and x-axis alongBOA. Let the coordinates ofAandBbe(a,0)and(−b,0). Let the fixed parallel linesAPandBQbe taken parallel to y-axis so thatBOAis perpendicular to both.
If∠POA=θthen as∠POQ=90o
We have∠QOB=90o−θ
Coordinates ofPare(a,atanθ)and those ofQare
(−b,bcotθ)
Equation to linePQis
y−atanθ=−(b+a)bcotθ−atanθ​(x−a)
orx(bcotθ−atanθ)+y(a+b)=ab(cotθ+tanθ)
or changing intosinθandcosθit becomes
x(bcos2θ−asin2θ)+y(a+b)sinθcosθ=ab
Multiplying by2, it becomes
x(2bcos2θ−2asin2θ)+y(a+b)sin2θ=2ab....(1)
Also equation to perpendicularORonPQis
x(a+b)sin2θ−y(2bcos2θ−2asin2θ)=0....(2)
But(2bcos2θ−2asin2θ)=b(1+cos2θ)−a(1+cos2θ)=(a+b)1+cos2θ−(a−b)
Hence the two lines (1) and (2) can be written as
x(a+b)cos2θ+y(a+b)sin2θ=2ab+(a−b)x.....(3)
andx(a+b)sin2θ−y(a+b)cos2θ=−(a−b)y.....(4)
In order to find the locus of point of intersectionR, we have to eliminateθfor which we square and add the equations (3) and (4)
∴x2(a+b)2+y2(a+b)2=4a2b2+4ab(a−b)x+(a−b)2(x2+y2)
or(x2+y2)((a+b)2−(a−b)2)=4ab(ab+(a−b)x)
x2−(a−b)x−ab+y2=0or(x−a)(y+b)+y2=0
above represents the equation of a circle onABas diameter.

Provide a better Answer & Earn Cool Goodies

Enter text here...
star
LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

tv

Full Live Access

material

Study Material

removal

Live Doubts Solving

assignment

Daily Class Assignments


Ask a Doubt

Get your questions answered by the expert for free

Enter text here...