Question

Let t be a variable quantity, find the locus of the point (x, y) when
$x=at+b{t}^2$ and $y=bt+a{t}^2.$

Answer 1

Given: $x=at+b{t}^2$ and $y=bt+a{t}^2$

To get the required locus, we are supposed to eliminate 't' from these equations.

$\Rightarrow$ $x=at+b{t}^2$ ..... $\left(i\right)$

$\Rightarrow$ $y=bt+a{t}^2$ ..... $\left(ii\right)$

Subtracting the eqn $b\times \left(ii\right)$ from $a\times \left(i\right)$, we get

$\Rightarrow$ $ax-by=(a^2-b^2)t$ ..... $I$

Subtracting eqn $b\times \left(i\right)$ from $a\times \left(ii\right)$ gives,

$\Rightarrow$ $ay-bx=(a^2-b^2)t^2$ ...... $II$

Substitute 't' from I in eqn. II

$\Rightarrow$ $ay-bx=(a^2-b^2)(\frac{ax-by}{a^2-b^2}{)}^2$

$\Rightarrow$ $(ax-by{)}^2=(a^2-b^2\left)\right(ay-bx)$

This is the required locus of the point (x,y).