Question

How do you find an equation of the tangent line to the parabola $y = {x^2} - 2x + 7\,$ at the point $(3,10)$ ?

Answer 1

Hint : First we will evaluate the slope of the tangent to the given curve that is a parabola. Then calculate where tangent we seek passes through the point $(3,10)$ . Then evaluate the equation of the tangent using slope point form of line.

Complete step-by-step answer :
The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point. The normal perpendicular to the tangent, so the product of their gradients is $- 1$ .
So now we will be differentiating $y = {x^2} - 2x + 7\,$ with respect to $x$ gives us:
$y = {x^2} - 2x + 7\,$
$\,\,\,y = {x^2} - 2x + 7\, \\\ \dfrac{{dy}}{{dx}} = 2x - 2 \;$
Now we will verify if the point $(3,10)$ lies on the curve.
Now we will substitute $x = 3$ in $y = {x^2} - 2x + 7\,$ .
$y = {x^2} - 2x + 7\, \\\ y = {3^2} - 2(3) + 7\, \\\ y = 9 - 6 + 7 \\\ y = 10\, \;$
Hence, the tangent we seek passes through $(3,10)$ and has the slope $4$ .
Now we will be using the slope point form and we will write the equation for the slope of the tangent.
The slope point form of the line is given by:
$(y - {y_1}) = m(x - {x_1})$
So, the required equation of tangent is,
${(y - 10)} = {4(x - 3)} \\\ {y - 10} = {4x - 12} \\\ y = {4x - 2}$
So, the correct answer is “ y = 4x - 2 ”.

Note : The slope of a line characterises the direction of a line. To find the slope, you divide the difference of the coordinates of two points on a line by the difference of the x-coordinates of those same two points.
Always calculate the slope of the line precisely. While calculating the slope of the line, always take into consideration the respective signs as well. Remember that the slope point form is represented by $(y - {y_1}) = m(x - {x_1})$ .