Question

The locus of the mid-point of the chords of the hyperbola `(x^2/a^2) - (y^2/b^2)` = 1 passing through a fixed point (α, β) is a hyperbola with centre at `(α/2, β/2)` It equation is ______.

Options

• `(x - α/2)^2/a^2 - (y - β/2)^2/b^2 = α^2/(4a^2) - β^2/(4b^2)`

• `(x + α/2)^2/a^2 - (y - β/2)^2/b^2 = α^2/(4a^2) + β^2/(4b^2)`

• `(x - α/2)^2/a^2 - (y - β/2)^2/b^2 = α/(4a^2) - β^2/(4b^2)`

• `(x + α/2)^2/a^2 - (y + β/2)^2/b^2 = α^2/(4a^2) β^2/(4b^2)`

Answer 1

The locus of the mid-point of the chords of the hyperbola `(x^2/a^2) - (y^2/b^2)` = 1 passing through a fixed point (α, β) is a hyperbola with centre at `(α/2, β/2)` It equation is `underlinebb((x - α/2)^2/a^2 - (y - β/2)^2/b^2 = α^2/(4a^2) - β^2/(4b^2))`.

Explanation:

Given equation of hyperbola `x^2/a^2 - y^2/b^2` = 1

Let the coordinates of the middle point of the chord be (h, k)

Applying T = S, will give equation of chord with mid-point as (h, k)

So `((xh)/a^2 - (yk)/b^2 - 1) = (h^2/a^2 - k^2/b^2 - 1)`

This line passes through (α, β) so we get

`(αh)/a^2 - (βk)/b^2 - 1 = h^2/a^2 - k^2/b^2 - 1`

⇒ `h^2/a^2 - (αh)/a^2 - k^2/b^2 + (βk)/b^2` = 0

⇒ `[h^2/a^2 - 2. h/a.α/(2α) + (α/(2α))^2] - [(k/b)^2 - 2. k/b + β/(2b) + (β/(2b))^2] = (α/(2a))^2 - (β/(2b))^2`

`("Hint: Adding" [(α/(2α))^2 - (β/(2b))^2]"on L.H.S and R.H.S")`

⇒ `(h/a - α/(2a))^2 - (k/b - β/(2b))^2 = α^2/(4a^2) - β^2/(4b^2)`

⇒ `(h - α/2)^2/a^2 - (k - β/2)^2/b^2 = α^2/(4a^2) - β^2/(4b^2)`

Hence the locus will be `(x - α/2)^2/a^2 - (y - β/2)^2/b^2 = α^2/(4a^2) - β^2/(4b^2)` with centre at `(α/2, β/2)`