\displaystyle 1.\ \text{If } y=f(x),\ \text{then}
\displaystyle \left(\frac{dy}{dx}\right)_P=\text{Slope of the tangent to } y=f(x)\ \text{at point } P.
\displaystyle -\frac{1}{\left(\frac{dy}{dx}\right)_P}=\text{Slope of the normal to } y=f(x)\ \text{at point } P.
\displaystyle \text{If the tangent is parallel to } x\text{-axis, then } \frac{dy}{dx}=0.
\displaystyle \text{If the tangent is parallel to } y\text{-axis, then } \frac{dx}{dy}=0.

\displaystyle 2.\ \text{If } P(x_1,y_1)\ \text{is a point on the curve } y=f(x),\ \text{then}
\displaystyle y-y_1=\left(\frac{dy}{dx}\right)_P(x-x_1)\ \text{is the equation of tangent at } P.
\displaystyle y-y_1=-\frac{1}{\left(\frac{dy}{dx}\right)_P}(x-x_1)\ \text{is the equation of the normal at } P.

\displaystyle 3.\ \text{The angle between the tangents to two given curves at their point of intersection is} \\ \text{defined as the angle of intersection of two curves.}
\displaystyle \text{If } C_1 \text{ and } C_2 \text{ are two curves having equations } y=f(x) \text{ and } y=g(x) \text{respectively such} \\ \text{that they intersect at point } P.\ \text{The angle } \theta \text{ of intersection }\text{of these two curves is given by}
\displaystyle \tan\theta=\frac{\left(\frac{dy}{dx}\right)_{C_1}-\left(\frac{dy}{dx}\right)_{C_2}}{1+\left(\frac{dy}{dx}\right)_{C_1}\left(\frac{dy}{dx}\right)_{C_2}}
\displaystyle \text{If the angle of intersection of two curves is a right angle, then the curves are said} \\ \text{to intersect orthogonally. The condition for orthogonality of two curves } C_1 \text{ and } C_2 \text{ is}
\displaystyle \left(\frac{dy}{dx}\right)_{C_1}\times\left(\frac{dy}{dx}\right)_{C_2}=-1

\displaystyle 4.\ \text{Two curves } ax^2+by^2=1 \text{ and } a'x^2+b'y^2=1 \text{ will intersect orthogonally, if}
\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{a'}+\frac{1}{b'}.

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