The number of possible tangents which can be drawn to the curve `4x^2-9y^2=36 ,` which are perpendicular to the straight line `5x+2y-10=0` , is zero (b) 1 (c) 2 (d) 4

Options

A. `5(y-3)=2(x-(sqrt(117))/(2)) `
B. ` 2x-5y+10-2sqrt(18)=0 `
C. ` 2x-5y-10-2sqrt(18)=0 `
D. none of these

Answer

D

Explanation

We have,
` 4x^(2)-9y^(2)=36 rArr 8x-18y(dy)/(dx)=0 rArr (dy)/(dx)=(4x)/(9y) `
` therefore " Slope of the tangent " =(4x)/(9y) `
For this tangent to be perpendicular to the straight line ` 5x+2y-10=0, ` we must have
`(4x)/(9y)xx (-(5)/(2))=-1 rArr y=(10x)/(9). `
Putting this value of y in `4x^(2)-9y^(2)=36, ` we get `-64x^(2)=324, ` which does not have real roots. Hence, at no point on the given curve can the tangent be perpendicular to the given line.