Question

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

Answer 1

Equation of a unit circle with centre at origin is $x^2+y^2=1$.

Given line, L: $2x+y=4$

Let $A(x_1,y_1)$ be a point on the line L.

Then $2x_1+y_1=4$

$\Rightarrow y_1=4-2x_1$

$QR\to$ with respect to point $A$.

The equation is $x{x}_1+y{y}_1=1..\left(i\right)$

$QR\to$ with respect to point $P(h,k)$

$\Rightarrow T=S_1$

$\Rightarrow hx+ky-1=h^2+k^2-1$

$\Rightarrow hx+ky=h^2+k^2..\left(ii\right)$

$\left(i\right)$ and $\left(ii\right)$represents the same line.

$\frac{x_1}{h}=\frac{y_1}{k}=\frac{1}{(h^2+k^2)}$

$\frac{x_1}{h}=\frac{4-2x_1}{k}=\frac{1}{(h^2+k^2)}...\left(iii\right)$

$k{x}_1=4h-2h{x}_1$

$(k+2h)x_1=4h$

$\frac{x_1}{h}=\frac{4}{(k+2h)}$

$\frac{x_1}{h}=\frac{1}{(h^2+k^2)}$ from $\left(iii\right)$

$\frac{4}{\left(k+2h\right)}=\frac{1}{(h^2+k^2)}$

$4(h^2+k^2)=k+2h$

$4(h^2+k^2)-k-2h=0$

Therefore, $4(x^2+y^2)-2x-y=0$ is the required equation.