Determine the coordinates of the point of intersection of the two perpendicular lines that intersect on the y-axis and are both tangent to the parabola given below.
y = 3x2
let y=mx+b be the first line
and y=1/m x+ d be the second line.
If y=3x^2 is tangent, then m=3, or m=1/3
.
y=3x^2=mx+b so m=3, b=0
y= 1/3 x +d, so m=1/3, d=0
oops. one of them has to be negative, so y=-1/3 x+ d, m=-1/3, d=0
Since the two tangents are mutually perpendicular and intersect on the y-axis they have gradients 1 and -1
The gradient function is y = 6x
Hence x = 1/6, -1/6
So y = 1/12 both times
So the coordinate of the intercept can be calculated using the equation of the tangents:
Equate y - 1/12 = x - 1/6
and y - 1/12 = -1(x + 1/6)
BUT you do not even have to do this!
Re-write one of the equations in the form
y = mx + c to get the y-intercept.
The coordinate required is
(0, -1/12)
To find the coordinates of the point of intersection of the two perpendicular lines, we need to determine the equations of the two lines first.
Given that the lines are tangent to the parabola and intersect on the y-axis, we can start by finding the equation of the tangent line to the parabola at a specific point (x, y).
The derivative of the parabola equation, y = 3x^2, will give us the slope of the tangent line at any point (x, y) on the parabola.
Let's find the derivative:
dy/dx = d/dx(3x^2)
= 6x
The slope of the tangent line at any point (x, y) on the parabola is determined by the value of 6x.
Next, we need to find the coordinates of the point on the parabola where the tangent line intersects the y-axis. Since the tangent line intersects the y-axis, the x-coordinate of this point is 0. By substituting x = 0 into the parabola equation, we can find the y-coordinate:
y = 3(0)^2
= 0
Therefore, the point of intersection of the tangent line and the y-axis is (0, 0).
Now, we can find the equation of the tangent line at this point, which is perpendicular to the y-axis. Since the y-axis is vertical, the tangent line will be horizontal and have a slope of 0.
Using the point-slope form of a line, where (x1, y1) is a point on the line and m is the slope, the equation of the tangent line is:
y - 0 = 0(x - 0)
y = 0
Therefore, the equation of the tangent line at the point (0, 0) is y = 0.
Since the tangent line is horizontal and the y-coordinate remains constant regardless of the x-coordinate, this line intersects the y-axis at y = 0.
Therefore, the coordinates of the point of intersection of the two perpendicular lines are (0, 0).