Question

If ax +by =p and bx-ay =q and a²+b²=1 then prove that p²+q²=x²+y²​

Answer 1

Answer:

Given that

ax +by =p ..........(1)

and

bx-ay =q ..........(2)

and

a²+b²=1 ..........(3)

And

we have to prove that

p²+q²=x²+y²

Squaring and adding (1) and (2)

We get

$(ax + by) {}^{2} + {(bx - ay)}^{2} = {p}^{2} + {q}^{2} \\ \\ {a}^{2} {x}^{2} + {b}^{2} {y}^{2} + \cancel{2axby} + {b}^{2} {x}^{2} + {a}^{2} {y}^{2} - \cancel{ 2bxay} = {p}^{2} + {q}^{2} \\ \\ ( {a}^{2} + {b}^{2} ) {x}^{2} + ( {b}^{2} + {a}^{2}) {y}^{2} = {p}^{2} + {q}^{2} \\ \\ (1) {x}^{2} + (1) {y}^{2} = {p}^{2} + {q}^{2} \: \: \: \: \: \: (\because \: using \: (3)) \\ \\ {x}^{2} + y {}^{2} = {p}^{2} + {q}^{2}$

Hence Proved