If `2x-y+1=0` is a tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(16)=1` then which of the following CANNOT be sides of a right angled triangle? `a ,4,2` (b) `a ,4,1` `2a ,4,1` (d) `2a ,8,1`

Options

A. 2a, 4, 1
B. 2a, 8, 1
C. a, 4, 1
D. a, 4, 2

Answer

B, C, D

Explanation

The line `y=mx+c` is tangent to hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^2)=1.`
if `c^(2)=a^(2)m^(2)-b^(2)`.
Given tangent is : `y=2x+1`
`therefore" "(1)^(2)=4a^(2)-16`
`rArr" "a^(2)=(17)/(4)`
`rArr" "a^(2)=(sqrt(17))/(2)`
For option (1), sides are `sqrt(17),4,1,` which form right angled triangle.
For option (2), sides are `sqrt(17),8,1,,` for which triangle is not possible.
For option (3), sides are `(sqrt(17))/(2),4,1,` for which triangle is not possible.
For option (4), sides are `(sqrt(17))/(2),4,2,` for which triangle exists but not right angled.