Question

A line $l$ has intercepts $a$ and $b$ on the coordinate axes . when the axes are rotated , the intercepts become $p$ & $q$ Find a relation between a, b, p & q ?

Answer 1

In the first coordinate system,

$\frac{x}{a}+\frac{y}{b}=1⟹\left(1\right)$

$\frac{x^{\prime }}{p}+\frac{y^{\prime }}{q}=1$

$x^{\prime }=xcos\theta +ysin\theta$

$y^{\prime }=-xsin\theta +ycos\theta$

so,$x\left(\frac{cos\theta }{p}-\frac{sin\theta }{q}\right)+y\left(\frac{sin\theta }{p}-\frac{cos\theta }{q}\right)=1$

comparing with (1),

$\frac{\left(\frac{\frac{cos\theta }{p}-\frac{sin\theta }{q}}{q}\right)}{\frac{1}{a}}=\frac{\left(\frac{\frac{sin\theta }{p}-\frac{cos\theta }{q}}{q}\right)}{\frac{1}{b}}=\frac{1}{1}$

$\frac{cos\theta }{p}-\frac{sin\theta }{q}=\frac{1}{a}⟹\left(2\right)$

$\frac{sin\theta }{p}-\frac{cos\theta }{q}=\frac{1}{b}⟹\left(3\right)$

Squaring and adding eqn 2 and 3

$\frac{1}{p^2}+\frac{1}{q^2}=\frac{1}{a^2}+\frac{1}{b^2}$