Equation of Tangent and Normal to the Curve

Here you will learn equation of tangent and normal to the curve with examples.

Let’s begin –

Equation of Tangent and Normal to the Curve

We know that the equation of line passing through a point \((x_1, y_1)\) and having slope m is \(y – y_1\) = m\((x – x_1)\).

and we know that the slopes of the tangent and the normal to the curve y = f(x) at a point P\((x_1, y_1)\) are \(({dy\over dx})_P\) and -\(1\over ({dy\over dx})_P\) respectively. 

Therefore the equation of the tangent at P\((x_1, y_1)\) to the curve y = f(x) is

\(y – y_1\) = \(({dy\over dx})_P\) (\(x – x_1\))

Since the normal at P\((x_1, y_1)\) passes through P and has slope -\(1\over ({dy\over dx})_P\).

Therefore, the equation of the normal at P\((x_1, y_1)\) to the curve y = f(x) is

\(y – y_1\) = \(-1\over ({dy\over dx})_P\) (\(x – x_1\))

Note :

1). If \(({dy\over dx})_P\) = \(\pm \infty\), then the tangent at \((x_1, y_1)\)  is parallel to y-axis and its equation is x = \(x_1\).

2). If \(({dy\over dx})_P\) = 0, then the normal at \((x_1, y_1)\)  is parallel to y-axis and its equation is x = \(x_1\).

3). If \(({dy\over dx})_P\) = \(\pm \infty\), then the normal at \((x_1, y_1)\)  is parallel to x-axis and its equation is y = \(y_1\).

4). If \(({dy\over dx})_P\) = 0, then the tangent at \((x_1, y_1)\)  is parallel to x-axis and its equation is y = \(y_1\).

Example : find the equation of the tangent to curve y = \(-5x^2 + 6x + 7\)  at the point (1/2. 35/4).

Solution : The equation of the given curve is 

y = \(-5x^2 + 6x + 7\) 

\(\implies\) \(dy\over dx\) = -10x + 6

\(\implies\) \(({dy\over dx})_{(1/2, 35/4)}\) = \(-10\over 4\) + 6 = 1

The required equation at (1/2, 35/4) is

y – \(35\over 4\) = \(({dy\over dx})_{(1/2, 35/4)}\) \((x – {1\over 2})\)

\(\implies\) y – 35/4 = 1(x – 1/2)

\(\implies\) y = x + 33/4

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