Question

The length of subnormal to the rectangular hyperbola $x^2-y^2=15$ at the point $(4,1)$ is ________

• A. $\frac{1}{4}$
• B. $4$
• C. $2$
• D. $8$

Answer 1

Answer: (B)

$4$

JK is the length of the subnormal .

From figure we can easily see that $J(4,0)$

Now, for $K$.

$x^2-y^2=15$

$⟹2x-2y\frac{dy}{dx}=0$

$⟹m=\frac{dy}{dx}=\frac{x}{y}=4$=slope of the tangent at P

Slope of normal that is $PK$ is $m_o=\frac{-1}{m}=\frac{-1}{4}$

Equation of line PK-

$y-1=\frac{-1}{4}(x-4)$

For $K$, $y=0$

$⟹x=8$

$⟹K(8,0)$

Now, $JK=4$=length of subnormal