Question

The line $y=4x+c$ touches the hyperbola $x^2-y^2=1$ if and only if

• A. $c=\pm \sqrt{2}$
• B. $c=\pm \sqrt{17}$
• C. $c=0$
• D. $c=\pm \sqrt{15}$

Answer 1

The correct option is D $c=\pm \sqrt{15}$

Given Hyperbola : $x^2-y^2=1$-------1 where $a=1,b=1$

Tangent: $y=4x+c$------------2 where $m=4,c=c$

For c, we have condition of tangency (i.e. the line equation 2 will touch hyperbola only when the following is true).

The condition of tangency $c^2=m^2{a}^2-b^2$--------------3

Putting the values of m,a,b in Equation 3, we get

$c^2=(4{)}^2\cdot (1{)}^2-(1{)}^2$

$c^2=16-1=\pm \sqrt{1}5$