Ellipse
11.0 Chord bisected at a given point
11.0 Chord bisected at a given point
The equation of a chord of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ bisected at the point $P({x_1},{y_1})$ can be find out using $T = {S_1}$ i.e., $$\frac{{x{x_1}}}{{{a^2}}} + \frac{{y{y_1}}}{{{b^2}}} - 1 = \frac{{{x_1}^2}}{{{a^2}}} + \frac{{{y_1}^2}}{{{b^2}}} - 1$$
Diameter of an ellipse is the locus of mid-point of all the chords of ellipse which are drawn parallel to each other
Let us find the locus of mid-point i.e., equation of chord when mid-point is known is given by $$T=S_1$$$$\frac{{x{x_1}}}{{{a^2}}} + \frac{{y{y_1}}}{{{b^2}}} = \frac{{{x_1}^2}}{{{a^2}}} + \frac{{{y_1}^2}}{{{b^2}}}$$ From the above equation, slope of chord $m$ which is constant is written as $$\begin{equation} \begin{aligned} \frac{{x{x_1}}}{{{a^2}}} + \frac{{y{y_1}}}{{{b^2}}} = \frac{{{x_1}^2}}{{{a^2}}} + \frac{{{y_1}^2}}{{{b^2}}} \\ m = - \frac{{{x_1}/{a^2}}}{{{y_1}/{b^2}}} \\\end{aligned} \end{equation} $$ We can write the equation of locus as $$\begin{equation} \begin{aligned} {y_1} = - \frac{{{b^2}}}{{m{a^2}}}{x_1} \\ \Rightarrow y = - \frac{{{b^2}}}{{m{a^2}}}x \\\end{aligned} \end{equation} $$
The conjugate diameter means the locus of mid-point of one chord drawn parallel to each other is the equation of another diameter of ellipse and vice-versa. For example, If $AB$ and $CD$ are two conjugate diameters, then the locus of mid-points of all chords drawn parallel to $CD$ represents the equation of $AB$.
