Here you will learn equation of tangent and normal to the curve with examples.
Letโs begin โ
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 \(y โ y_1\) = m\((x โ x_1)\).
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.ย
Therefore the equation of the tangent at P\((x_1, y_1)\) to the curve y = f(x) is
\(y โ y_1\) = \(({dy\over dx})_P\) (\(x โ x_1\))
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
\(y โ y_1\) = \(-1\over ({dy\over dx})_P\) (\(x โ x_1\))
Note :
1). If \(({dy\over dx})_P\) = \(\pm \infty\), then the tangent at \((x_1, y_1)\)ย is parallel to y-axis and its equation is x = \(x_1\).
2). If \(({dy\over dx})_P\) = 0, then the normal at \((x_1, y_1)\)ย is parallel to y-axis and its equation is x = \(x_1\).
3). If \(({dy\over dx})_P\) = \(\pm \infty\), then the normal at \((x_1, y_1)\)ย is parallel to x-axis and its equation is y = \(y_1\).
4). If \(({dy\over dx})_P\) = 0, then the tangent at \((x_1, y_1)\)ย is parallel to x-axis and its equation is y = \(y_1\).
Example : find the equation of the tangent to curve y = \(-5x^2 + 6x + 7\)ย at the point (1/2, 35/4).
Solution : The equation of the given curve isย
y = \(-5x^2 + 6x + 7\)ย
\(\implies\) \(dy\over dx\) = -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 โ \(35\over 4\) = \(({dy\over dx})_{(1/2, 35/4)}\) \((x โ {1\over 2})\)
\(\implies\) y โ 35/4 = 1(x โ 1/2)
\(\implies\) y = x + 33/4
Related Questions
Find the equation of the normal to the curve y = \(2x^2 + 3 sin x\) at x = 0.
