The coordinates of the point on the parabola y = x 2 + 7 x + 2 , which is nearest to the straight line y = 3 x − 3 are

Question

TestHub · Accepted Answer

Step 1: Determine the required slope

The point on the parabola nearest to the line

$y = 3 x - 3$

has a tangent line parallel to the given line. The slope of the line

$y = 3 x - 3$

is

$m = 3$ .

Step 2: Find the derivative of the parabola

The equation for the parabola is

$y = x^{2} + 7 x + 2$ .

The derivative

$\frac{d y}{d x}$

gives the slope of the tangent at any point

$(x, y)$ :

$\frac{d y}{d x} = 2 x + 7$

Step 3: Solve for the x-coordinate

Set the derivative (slope of the tangent) equal to the slope of the given line to find the specific

$x$ -coordinate:

$2 x + 7 = 3$

$2 x = - 4$

$x = - 2$

Step 4: Calculate the y-coordinate

Substitute the value of

$x = - 2$

back into the original parabola equation to find the corresponding

$y$ -coordinate:

$y = (- 2)^{2} + 7 (- 2) + 2$

$y = 4 - 14 + 2$

Mathematics - Parabola Question with Solution | TestHub