Let l be a line which is normal to the curve \(y = 2x^2 + x + 2\) at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to _________ .
Correct Answer: 13
Solution and Explanation
\(\frac {y_1 - 4}{ x_1 - 6} = - \frac {1}{4x_1+1}\)
\(⇒\frac { 2x^2_1 + x_1 - 2}{x_1 - 6} = - \frac {1}{4x_1+1}\)
\(= 6 - x_1 = 8x_1^3 + 6x_1^2 - 7x_1 - 2\)
\(⇒ 8x_1^3 + 6x_1^2 – 6x_1 – 8 = 0\)
So, \(x_1 = 1 ⇒y_1 = 5\)
Area = \(\frac 12 \begin{vmatrix} 0 & 0 & 1 \\[0.3em] 6 & 4 & 1 \\[0.3em] 1 & 5 & 1 \end{vmatrix}\)
\(= 13\)
Hence, the answer is \(13\).
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Concepts Used:
Tangents and Normals
- A tangent at a degree on the curve could be a straight line that touches the curve at that time and whose slope is up to the derivative of the curve at that point. From the definition, you'll be able to deduce the way to realize the equation of the tangent to the curve at any point.
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m×n = -1
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Diagram Explaining Tangents and Normal:
