Question

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

• A. $2(x^2+y^2)=x+y$
• B. $2(x^2+y^2)=x+2y$
• C. $4(x^2+y^2)=2x+y$
• D. None of these

Answer 1

The correct option is C $4(x^2+y^2)=2x+y$

$x^2+y^2=1$

$L:2x+y=4$

$2x_1+y_1=4$

$y_1=4-2x_1$

$QK\to$ withy point $A(x_1,y_1)$

$x{x}_1+y{y}_1=1$ ......... $I$

$QK\to$ with given middle points $p(h,k)$

$T\Rightarrow S_1$

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

$hx+ky=h^2+k^2$ .......... $II$

$I$ & $II$ represent the same line

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

$\frac{x_1}{h}=\frac{4-2x_1}{k}=\frac{1}{h^2+k^2}$

$k{x}_1=4h-2h\times x_1$

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

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

$\frac{x_1}{h}=\frac{h}{h^2+k^2}$

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

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

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

$\Rightarrow 4(x^2+y^2)-2x-y=0$