Question

If the tangents drawn to the hyperbola 4y2=x2+1 intersect the co-ordinate axes at the distinct points a and b, then the locus of the mid point of ab is

Answer 1

equation of hyperbola is 4y² = x² + 1
-x² + 4y² = 1
- x²/1 + y²/(1/2)² = 1
compare this equation with standard equation of hyperbola, y²/b² - x²/a² = 1
so, a = 1 and b = 1/2

now tangent to the curve at a point (x1, y1) is given by,
-2x + 8y dy/dx = 0
-2x1 + 8y1 dy/dx = 0
dy/dx = x1/4y1

equation of tangent at (x1, y1) is
(y - y1) = x1/4y1(x - x1)
or, 4y1(y - y1) = x1(x - x1)
or, 4yy1 - 4y1² = xx1 - x1²
or, xx1 - 4yy1 = 4y1² - x1²

tangent intersects the co-ordinate axes at the distinct points a and b.
so, a(-1/x1, 0) and b(0, 1/4y1)
let locus of midpoint of line of tangent is (h, k)
so, h = - 1/2x1 => x1 = -1/2h
and k= 1/8y1 => y1 = 1/8k

Thus, 4(1/8k)² = (-1/2h)² + 1

4/64k² = 1/4h² + 1

1/16k² = 1/4h² + 1

hence, 1/16y² = 1/4x² + 1 is the locus of midpoint of tangent ab