Correct Answer - Option 1 : 6y + x = 85 and 6x – y = 15
Concept:
Equation of tangent is \(\frac{{y\; - \;f\left( a \right)}}{{x\; - \;a}} = f'\left( a \right)\) for the function y = f(x) at x = a
Since tangent and normal are perpendicular to each other, product of slope of the tangent and slope of the normal will be equal to -1.
Equation of the normal will be \(\frac{{y\; - \;f\left( a \right)}}{{x\; - \;a}} = \frac{{ - 1}}{{f'\left( a \right){\rm{\;}}}}\)
Calculation:
Given: f(x) = x2 – 4x + 10
⇒ f(5) = 15
⇒ f’(x) = 2x – 4
⇒ f’(5) = 6
As we know that equation of tangent is \(\frac{{y\; - \;f\left( a \right)}}{{x\; - \;a}} = f'\left( a \right)\) for the function y = f(x) at x = a
\(\frac{{y - 15}}{{x - 5}} = 6\)
⇒ (y – 15) = 6 (x – 5)
⇒ 6x – y – 15 = 0
Equation of tangent is 6x – y =15
Since tangent and normal are perpendicular to each other, product of slope of the tangent and slope of the normal will be equal to -1.
Slope of normal = - 1/f’(5) = - 1/6
As we know that equation of the normal will be \(\frac{{y\; - \;f\left( a \right)}}{{x\; - \;a}} = \frac{{ - 1}}{{f'\left( a \right){\rm{\;}}}}\)
\(\frac{{y - 15}}{{x - 5}} = \; - \frac{1}{6}\)
⇒ 6 (y – 15) = 5 - x
Equation of normal is 6y + x = 85
