Question

The equation of the normal to the hyperbola $x^2-9y^2=7$ at the point (4,1) is .

• A. 9x + 4y = 40
• B. 9x - 4y = 40
• C. 4x + 9y = 40
• D. 4x - 9y = 40

Answer 1

The correct option is A 9x + 4y = 40

$\mathrm{Differentiating}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{hyperbola}$
$x^2-9y^2=7$ $w.r.tx,\mathrm{we}\mathrm{get}-$ $2x-18y\frac{\mathrm{dy}}{\mathrm{dx}}=0\mathrm{or}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{x}{9y}\mathrm{Now}\mathrm{the}\mathrm{slope}\mathrm{of}a\mathrm{normal}\mathrm{at}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{curve}\mathrm{is}-\frac{\mathrm{dx}}{\mathrm{dy}}=-\frac{9y}{x}\mathrm{Hence}\mathrm{the}\mathrm{slope}\mathrm{of}\mathrm{the}\mathrm{normal}\mathrm{at}(4,1)\mathrm{is}-\frac{9}{4}\mathrm{Hence},\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{normal}\mathrm{is}:y-1=-\frac{9}{4}(x-4)\mathrm{or}9x+4y=40$