A variable rectangle PQRS has its sides parallel to fixed directions Q S lie respectively on the - Maths - Straight Lines

Question

A variable rectangle PQRS has its sides parallel to fixed
directions Q & S lie respectively on the line x = a, x = - a
and P lies on the X axis Prove that Locus of R is a line

Rahul Raj · Accepted Answer

dear student

Let Q be (a,b) and S be (-a,c), where b and c are variables

Let R be (h,k)

as P lies on x-axis let P be (z,0)midpoint of PR=(h+z2,k+02)=(h+z2,k2)midpoint of QS=(a-a2,b+c2)=(0,b+c2)as midpoint is samez=-h, and k=b+cSo, P is (-h,0)PQ and PS are perpendicular ba+h×c(-a+h)=-1bc=a2-h2further sides of rectangle are parallel to fixed directionssoba+h=m (constant) then b=m(a+h)thenc-a+h=-1m(constant) gives c=a-hmk=b+c=m(a+h)+a-hmk=m2a+m2h+a-hmm2a+m2h+a-h=mkm2a+m2h+a-h-mk=0(m2-1)h-mk+a(1+m2)=0so locus is (m2-1)x-my+a(1+m2)=0which is a line

regards

A variable rectangle PQRS has its sides parallel to fixed directions. Q & S lie respectively on the line x = a, x = - a and P lies on the X axis. Prove that Locus of R is a line.