Multiple choice

The locus of the mid-point of the chords of x2/a2 -y2/b2 =1, which pass through a fixed point P(x1, y1) is

  1. A circle

  2. An ellipse

  3. A hyperbola

  4. None of these
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C Correct answer
Explanation

For a hyperbola x^2/a^2 - y^2/b^2 = 1, the locus of the midpoint (h, k) of a chord passing through (x1, y1) is given by T = S1, where T is the equation of the chord and S1 is the value of the hyperbola at the midpoint. This results in a hyperbola.

AI explanation

Let the midpoint of a chord be (h, k). Using the midpoint chord formula for a hyperbola, the equation of the chord is (h times x divided by a squared) minus (k times y divided by b squared) equals (h squared divided by a squared) minus (k squared divided by b squared). Because this chord passes through the fixed point P(x1, y1), we substitute these coordinates to get (hx1 divided by a squared) minus (ky1 divided by b squared) equals (h squared divided by a squared) minus (k squared divided by b squared). Replacing h and k with x and y reveals that the resulting equation is a hyperbola.

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