Question: If chords of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ pass through a fixed poin...

Question

If chords of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ pass through a fixed point (h, k) then the locus of their middle points is a

Answer 1

Let (x₁, y₁) be the mid point of any chord equation of the chord having (x₁, y₁) as its mid point is

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

If it passes through the fixed point (h, k) then

$\frac{hx_{1}}{a^{2}} + \frac{ky_{1}}{b^{2}} = \frac{x_{1}^{2}}{a^{2}} + \frac{y_{1}^{2}}{b^{2}}$ .

∴ Locus of (x₁, y₁) is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{hx}{a^{2}} - \frac{ky}{b^{2}} = 0$ , which is

another ellipse.

Question: If chords of the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) pass through a fixed poin...