Correct Answer - Option 1 : 2x - y = 0
Concept:
To determine the equation of a tangent to a curve:
Step. 1) Find the derivative of given curve
Step. 2) Calculate the gradient of the tangent at given point.
Step. 3) Determine the equation of tangent.
Substitute the gradient of the tangent and the coordinates of the given point into the gradient-point form of the straight line equation \(\rm (y - y_1) = m (x - x_1)\).
Calculations:
Given equation of curve is y = x2 + 4x + 1.
tep. 1) Find the derivative of given curve
Differentiate w.r.to x on both side, we get
\(\rm\dfrac{dy}{dx}= m = 2x + 4\)
Step. 2) Calculate the gradient of the tangent at given point
To determine the gradient of the tangent at the point (-1, -2), put x = -1 into the equation for the derivative.
⇒ m =2 (- 1) + 4
⇒ m = 2
Step. 3) Determine the equation of tangent.
Substitute the gradient of the tangent and the coordinates of the given point into the gradient-point form of the straight line equation.
\(\rm (y - y_1) = m (x - x_1)\)
⇒ (y +2) = 2(x+ 1)
⇒ 2x - y = 0
Hence, the equation of tangent to the curve y = x2 + 4x + 1 at (-1, -2) is 2x - y = 0
