Question

If the tangent at point P(h, k) on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` cuts the circle `x^(2)+y^(2)=a^(2)` at points `Q(x_(1),y_(1))` and `R(x_(2),y_(2))`, then the vlaue of `(1)/(y_(1))+(1)/(y_(2))` is
A. `(1)/(k)`
B. `(2)/(k)`
C. `(ab)/(k)`
D. `(a+b)/(k)`

Answer 1

Correct Answer - B
Equation of tangent at P is
`(xh)/(a^(2))-(yk)/(b^(2))=1`
`therefore" "x=(1+(yk)/(b^(2)))(a^(2))/(h)`
Putting into the equation `x^(2)+y^(2)=a^(2)`, we get
`(1+(y^(2)k^(2))/(b^(4))+(2yk)/(b^(2)))(a^(4))/(h^(2))+h^(2)=a^(2)`
`rArr" "y^(2)((k^(2)a^(4))/(b^(4)h^(2))+1)+(2ka^(4))/(b^(2)h^(2))y+((a^(4))/(h^(2))-a^(2))=0`
`rArr" "(1)/(y_(1))+(1)/(y_(2))=(y_(1)y_(2))/(y_(1)y_(2))`
`=((-2ka^(4))/(b^(2)h^(2)))/((a^(4)-a^(2)h^(2))/(h^(2)))=-(2ka^(2))/(b^(2)(a^(2)-h^(2)))`
`=-(2k)/(b^(2)(1-(h^(2))/(a^(2))))=-(2k)/(b^(2)(-(k^(2))/(b^(2))))=(2)/(k)`