Question

The locus of the mid-points of a system of parallel chords of an ellipse is a

• A. Straight line
• B. Parabola
• C. Pair of straight lines
• D. None of these

Answer 1

Answer: (A)

Straight line

Answer 2

Let m be the slope of the system of parallel chords of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$. Let M(x₁, y₁) be the mid point of any one chord of the system. Then its equation is

T = S₁ i.e. $\frac{xx_{1}}{a^{2}} + \frac{yy_{1}}{b^{2}} - 1 = \frac{x_{1}^{2}}{a^{2}} + \frac{y_{1}^{2}}{b^{2}} - 1$.

Its slope is = $\frac{- b^{2}x_{1}}{a^{2}y_{1}}$.

But the slope is given to be m.

∴ $\frac{- b^{2}x_{1}}{a^{2}y_{1}} = m \Rightarrow y_{1} = \frac{- b^{2}}{a^{2}m}x_{1}$.

Hence locus of (x₁, y₁) is y = $\frac{- b^{2}}{a^{2}m}x$, which is a straight line through the centre (0, 0) of the ellipse.