The line y = hx - 2 where h > 0 meets the curve 3y² = 9x² - 6x + 1.

Mathematics

1 answer:

I am Lyosha [343]2 years ago

8 0

Answer:

h=6

Step-by-step explanation:

since $y =hx -2$ is an equation for a line which intersects with the curve $3y^2 = 9x^2 - 6x +1$ . The point of intersection, let's say $(x_1,y_1)$ , should satisfy the two equations. As a result, the value of y in the second equation can be replaced with the value of y in the first equation as the following,

$3y^2 = 3(hx-2)^2 = 9x^2 - 6x +1\\3(h^2x^2 -4hx + 4) = 9x^2 - 6x + 1\\3h^2 x^2 -12hx +12 = 9x^2 - 6x +1$

therefore, the latter equation can be rewritten in a quadratic equation form as the following,

$(9 - 3h^2) x^2 + (12h-6)x + (1-12) = (9 - 3h^2) x^2 + (12h-6)x -11$ = 0

if the line is tangent to the curve, it means that the line touches the curve at one point, therefore the discernment of the second order equation will be equal to zero for the famous quadratic equation solution.

$b^2 -4ac =0$

where $a=9-3h^2, b=12h-6$ and $c=-11$ , as a result, the following equations can be deduced,

$(12h-6)^2 - 4*(9-3h^2)*(-11) = (144h^2 -144h +36) - (36 -12h^2)*(-11)=\\(144-(-12)*(-11)h^2 ) -144h +36 -(36*(-11)) =\\12 h^2 -144h +432 =0$

therefore, dividing both sides by 12

$h^2 -12 +36 =0\\(h-6)^2 =0\\h=6$

You might be interested in

Points (−6, 3) and (−6, −3) on the coordinate grid below show the positions of two dogs at a park: A coordinate plane is shown.

Nostrana [21]

Question:

Points (−6, 3) and (−6, −3) on the coordinate grid below show the positions of two dogs at a park.There is a point at (−6, 3) labeled Dog 1. There is a point at (−6, −3) labeled Dog 2.

Which statement best describes the relationship between the positions of the two players?

A.Dog 1's position is Dog 2's position reflected across the y-axis; only the signs of the y-coordinates of Dog 1 and Dog 2 are different.

B.Dog 1's position is Dog 2's position reflected across the x-axis; only the signs of the y-coordinates of Dog 1 and Dog 2 are different.

C.Dog 1's position is Dog 2's position reflected across the x-axis; only the signs of the x-coordinates of Dog 1 and Dog 2 are different.

D. Dog 1's position is Dog 2's position reflected across the y-axis; only the signs of the x-coordinates of Dog 1 and Dog 2 are different.

Answer:

Option B: Dog 1's position is Dog 2's position reflected across the x-axis; only the signs of the y-coordinates of Dog 1 and Dog 2 are different.

Explanation:

The position of the Dog 1 is (−6, 3).

The position of the Dog 2 is (−6, -3).

From the graph, we can see that the image of Dog 1 is reflected over x-axis. Because to reflect an image over x-axis, the signs of y-coordinate must be changed.

Hence, Option B is correct.

Thus, Dog 1's position is Dog 2's position reflected across the x-axis; only the signs of the y-coordinates of Dog 1 and Dog 2 are different.