Question

The equation of a line perpendicular to the
tangent to the hyperbola
$xy=16\text{at the point}(4,4)$
and passing through the intercept made by the tangent on the x-axis is

• A. $x+y=4$
• B. $x+4=y$
• C. $y+4=x$
• D. $y+4=x$

Answer 1

The correct option is D $y+4=x$

Given, $xy=16,$
Equation of tangent on hyperbola
$xy=c^2\text{at the point}P(x_1,y_1)\text{is}$
$x{y}_1+y{x}_1=c^2$
$\text{Point}:(x_1,y_1)\equiv (4,4)$
$\Rightarrow x\times 4+y\times 4=16$
$\Rightarrow x+y=4$
$\text{Slope of tangent}=-1$
X-intercept:
$\text{At}y=0\Rightarrow x=4$
Point: $\equiv (4,0)$

The slope of a line perpendicular to the tangent is $1.$
Hence, equation:
$\Rightarrow y-0=1(x-4)$
$\Rightarrow y+4=x$
Hence, the correct answer is Option b.