A variable chord of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1,(b > a),` subtends a right angle at the center of the hyperbola if this chord touches. a fixed circle concentric with the hyperbola a fixed ellipse concentric with the hyperbola a fixed hyperbola concentric with the hyperbola a fixed parabola having vertex at (0, 0).

Options

A. a fixed circle concentric with the hyperbola
B. a fixed ellipse concentric with the hyperbola
C. a fixed hyperbola concentric with the hyperbola
D. a fixed parabola having vertex at (0, 0)

Answer

A

Explanation

Let the variable chord be
`x cos alpha+y sin alpha=p" (1)"`
Let this chord intersect the hyperbola at A and B. Then the combined equation of OA and OB is given by
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=((x cos alpha+y sin alpha)/(p))^(2)`
`x^(2)((1)/(a^(2))-(cos^(2)alpha)/(p^(2)))-y^(2)((1)/(b^(2))+(sin^(2)alpha)/(p^(2)))-(2 sin alpha cos alpha)/(p)xy=0`
This chord subtends a right angle at the center. Therefore,
`"Coefficient of " x^(2)+"Coefficient of "y^(2)=0`
`"or "(1)/(a^(2))-(cos^(2)alpha)/(p^(2))-(1)/(b^(2))-(sin^(2) alpha)/(po)=0`
`"or "(1)/(a^(2))-(1)/(b^(2))=(1)/(p^(2))`
`"or "p^(2)=(a^(2)b^(2))/(b^(2)-a^(2))`
Hence, p is constant, i.e., it touches the fixed circle.