Question

If $(x_1,y_1)$ is a point inside the circle $x^2+y^2+2gx+2fy+c=0$ Given two expressions $S_1$ and $T_1$ such that,

$S_1=x_1^2+{y_1}^2+2g{x}_1+2f{y}_1+c$

$T_1=x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c$

Then the equation of chord centered at $(x_1,y_1)$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

We are again making use of the fact here that a chord is always perpendicular to the radius going through its midpoint. The circle is given by the following figure.

In the figure $AB\perp OP$

$\because SlopeofAB=\frac{-1}{slopeofOP}$

$=\frac{-1}{\frac{(y_2+f)}{(x_1+g)}}$

$-\frac{(x_1+g)}{(y_2+f)}$

$\therefore$ Equation of the given chord=Equation a line passing through $(x_1,y_1)$ and having slope of $-\frac{(x_1+g)}{(y_1+f)}$

$\therefore y-y_1=-\frac{(x_1+g)}{(y_1+f)}.(x-x_1)$

$(y-y_1)(y_1+f)+(x_1+g)(x-x_1)=0$

$y{y}_1+y_1^2+fy-f{y}_1+x_{1}x-x_1^2+gx-g{x}_1=0$

$x_1^2+y_1^2+f{y}_1+g{x}_1=y{y}_1+x_{1}x+fy+gx$

adding $f{y}_1+g{x}_1+c$ to both sides

$x{x}_1+y{y}_1+f(y+y_1)+g(x+x_1)+c=y_1^2+x_1^2+2g{x}_1+2f{y}_1+c$

$\Rightarrow T_1=S_1$

$\Rightarrow T_1-S_1=0$

This can be used as a formula for finding the equation of chord with given midpoint.