Tangents are drawn to a unit circle with centre at the origin from each point on the line $2\mathrm{x}+\mathrm{y}=4.$ Then the equation to the locus of the middle point of the chord of contact is

$2{\mathrm{x}}_1+{\mathrm{y}}_1=4 .......\left(1\right)$

chord of cotntact w.r.t. (x₁ , y₁)

${\mathrm{xx}}_1+{\mathrm{yy}}_1=1$ 

also equation of chord

whose mid point is $(\mathrm{h},\mathrm{k}){\mathrm{h}}^2+{\mathrm{k}}^2=\mathrm{hx}+\mathrm{ky}$

$\Rightarrow \frac{{\mathrm{x}}_1}{\mathrm{h}}=\frac{{\mathrm{y}}_1}{\mathrm{k}}=\frac{1}{{\mathrm{h}}^2+{\mathrm{k}}^2}\Rightarrow {\mathrm{x}}_1=\frac{\mathrm{h}}{{\mathrm{h}}^2+{\mathrm{k}}^2};{\mathrm{y}}_1=\frac{\mathrm{k}}{{\mathrm{h}}^2+{\mathrm{k}}^2}$

substitute in (1) $2\frac{\mathrm{h}}{{\mathrm{k}}^2+{\mathrm{k}}^2}+\frac{\mathrm{k}}{{\mathrm{h}}^2+{\mathrm{k}}^2}=4$

$\text{ locus }=4\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)=2\mathrm{x}+\mathrm{y}$