Sketch a graph of the parabola y=x^2+3. On the same graph, plot the point (0,−6). Note there are two tangent lines of y=x2+3 that pass through the point (0,−6).
The tangent line of the parabola y=x^2+3 at the point (a,a^2+3) passes through the point (0,−6) where a>0. The other tangent line that passes through the point (0,−6) occurs at the point (−a,a^2+3).
Find the number a
The slope at any point (x,y) is y'(x) = 2x
So, we want the line
y-(a^2+3) = 2a(x-a)
to pass through (0,-6)
-6-a^2-3 = -2a^2
a^2 - 9 = 0
a = 3
So, the lines tangent to x^2+3 at (3,12) and (-3,12) both pass through (0,-6)
The slopes of the tangent lines are equal to the derivative of the parabola at the points. y ' = 2x, so y'(a) = 2a and y'(-a) = -2a.
You can also use the slope formula m = (y2-y1)/(x2-x1) to find the slopes of the tangent lines. Therefore m = (a^2+3-(-6))/(a-0)=(a^2+9)/a. Setting this equal to 2a, and solving for a, you get a = 3 and a = -3
To find the value of a, we need to set up the equation for the tangent line of the parabola y = x^2 + 3 at the point (a, a^2 + 3).
The equation of a tangent line to a curve can be expressed in the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the curve and m is the slope of the tangent line at that point.
First, we need to find the slope of the tangent line. The derivative of the function y = x^2 + 3 will give us the slope at any point on the parabola.
Taking the derivative of y = x^2 + 3, we get: dy/dx = 2x.
Evaluating the derivative at the point (a, a^2 + 3), we have m = 2a.
Now we can plug in the values of the point (a, a^2 + 3) and the slope 2a into the point-slope form equation to get the equation of the tangent line:
y - (a^2 + 3) = 2a(x - a)
Simplifying this equation, we have:
y = 2ax - 2a^2 + a^2 + 3
y = 2ax - a^2 + 3
Since this tangent line passes through the point (0, -6), we can substitute these coordinates into the equation:
-6 = 2a(0) - a^2 + 3
-6 = -a^2 + 3
Rearranging the equation:
a^2 = 9
Taking the square root of both sides:
a = ±3
Therefore, there are two values of a that satisfy the given conditions: a = 3 and a = -3.