Question

The curves $4x^2+9y^2=72$ and $x^2-y^2=5$ at (3, 2):

• A. touch each other
• B. cut orthogonally
• C. intersect at $45{}^{\circ }$
• D. intersect at $60{}^{\circ }$
• E. none of these

Answer 1

Answer: (B)

The given curves are $4x^2+9y^2=72$ ...(i) and $x^2-y^2=5$ ?(ii) On differentiating Eq. (i) w. r. t. $x,$ we get $8x+18y\frac{dy}{dx}=0$ $\frac{dy}{dx}=-\frac{4x}{9y}$ $\therefore$ Slope of Eq.(i) $=m_1={\left(\frac{dy}{dx}\right)}_{(3,2)}=-\frac{2}{3}$ On differenting Eq. (ii) w. r. t. $x,$ we get $2x-2y\frac{dy}{dx}=0$ $\Rightarrow$ $\frac{dy}{dx}=\frac{x}{y}$ $\therefore$ Slope of Eq.(ii) $=m^2={\left(\frac{dy}{dx}\right)}_{(3,2)}=\frac{3}{2}$ $\therefore$ $m_1{m}_2=\frac{-2}{3}\times \frac{3}{2}=-1$ $\therefore$ Both the curves cut orthogonally.