about to only mathematics

Options

A. `((9)/(2sqrt(2)),(1)/(sqrt(2)))`
B. `(-(9)/(2sqrt(2)),-(1)/(sqrt(2)))`
C. `(3sqrt(3),2sqrt(2))`
D. `(-3sqrt(3),-2sqrt(2))`

Explanation

PLAN Equation of tangent to `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1 " is " y=mx pm sqrt(a^(2)m^(2)-b^(2))`
Description of Situation If two straight lines
`a_(1)x+b_(1)y+c_(1)=0`
and `a_(2)x+b_(2)y+c_(2)=0 " are identical. Then, " (a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))`
Equation fo tangent, parallel to `y=2x-1` is
`y= 2x pm sqrt(9(4)-4)`
`therefore y=2x pm sqrt(32) " ...(i)" `
The equation of tanent at `(x_(1),y_(1))` is
`(x x_(1))/(9)-(y y_(1))/(4)=1`
From Eqs. (i) and (ii),

`(2)/((x_(1))/(9))=(-1)/((-y_(1))/(4))=(pmsqrt(32))/(1)`
` x_(1)=-(9)/(2sqrt(2))and y_(1)= -(1)/(sqrt(2))`
`or x_(1)=(9)/(2sqrt(2)),y_(1)=(1)/(sqrt(2))`