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A chord PQ of the hyperbola `xy=c^2` is tangent to the hyperbola `x^2 / a^2 - y^2 / b^2 = 1` Find the locus of the middle point of PQ. |
| Answer» Given, hyperbola `xy = c^2`Let its chord `PQ` has the mid point `(h,k)``Eqn.` of chord with a given point`T = S_1`From eqn. of hyperbola, tangent:`T= 2*xy/2 = c^2``rArr T = (xy + xy)/2 - c^2`At point `(h,k)``T = (hy + kx)/2 - c^2``S_1` refers to point satisfying eqn. of hyperbola.`S_1 = hk - c^2``:. (hy+kx)/2 - c^2 = hk - c^2``rArr kx + hy = 2hk``hy = 2hk - kx``y = (2hk - kx)/h` ..... (from `eq.1`)This is the chord which is tangent to another hyperbola:`x^/a^2 - y^2/b^2 = -1`Applying condition of tangencySatisfying eq. of hyperbola and eq. of tangent.`x^2/a^2 - (2hk - kx)^2/(h^2*y^2) = -1` (From `eq. 1`)Simplifying`x^2/a^2 - (4h^2*k^2 + k^2*x^2 - 4h*k^2*x)/h^2*b^2 = -1``rArr x^2/a^2 - (4h^2*k^2)/(h^2*b^2) - (k^2*x^2)/(h^2*b^2) + (4h*k^2*x)/(h^2*b^2) + 1 = 0``rArr (1/a^2 - k^2/(h*b^2))*x^2 + (4*k^2/(h*b^2))*x + ( 1 - 4*k^2/b^2)`This must have equal roots and `D = 0``D = (4k^2/(h*b^2))^2 - 4*(1/a^2 - k^2/(h^2*b^2))(1 - 4k^2/b^2) = 0``rArr 4k^4/(h^2*b^4) - (1/a^2 - 4*k^2/(a^2.b^2) - k^2/(h^2*b^2) + 4k^4/(h^2*b^4)) = 0``rArr 4k^4/(h^2*b^4) - 1/a^2 + 4*k^2/(a^2.b^2) + k^2/(h^2*b^2) - 4k^4/(h^2*b^4) = 0``rArr 4k^2/(a^2*b^2) + k^2/(h^2*b^2) = 1/a^2`Required locus:`4y^2/(a^2*b^2) + y^2/(x^2*b^2) = 1/a^2` | |