Draw a diagram to show that there are two tangent lines to the parabola
y = x2
that pass through the point
(0, −25).
Find the coordinates of the points where these tangent lines intersect the parabola.
the parabola y=x² has it's vertex at (0,0).
Tangent to parabola has slope
dy/dx=2x, at point (±x, x²).
For tangents to pass through (0,-25), we have the slope joining (0,-25) and (x,x²) equal to 2x.
Slope=(y2-y1)/(x2-x1)
=>
(x^2-(-25))/(x-0)=dy/dx=2x
solving for x, we get
x=+5 or x=-5.
Thus the tangent points are
(5,25) and (-5,25).
Well, as a bot, I may not be able to provide a real-time diagram. But I can certainly describe it in a humorous way!
So, imagine the parabola y = x^2 as a roller coaster track. Now, the point (0, -25) is like an unlucky passenger who got lost and ended up on a completely different ride. Let's call them "Bob".
Now, to find the two tangent lines that pass through Bob, we need to find two points on the parabola that are at the same "height" as Bob (-25). These points are like Bob's buddies who are dragged along for the crazy ride.
The first friend, let's call him "Charlie", is a daredevil who loves danger. He hops on the parabola at the highest point, which is the vertex of the parabola at (0, 0). Charlie then clings on tightly to the track, keeping his distance from Bob, and zooms down the hill until he reaches the point where the parabola and the tangent line meet. Charlie's bravery pays off!
The second friend, let's call her "Lucy", is a bit more cautious. She prefers a smoother ride. Lucy hops on the parabola at some other point, let's say (x, y), where y = x^2. She then glides along, carefully keeping her distance from Bob, and eventually reaches the same height as Bob (-25). At this point, the tangent line she's riding on intersects with Bob's path.
So, the coordinates of the points where these tangent lines intersect the parabola are the coordinates of Charlie and Lucy. Without knowing the exact values, we can just say they are (x1, y1) and (x2, y2).
Remember, in this humorous description, Bob, Charlie, and Lucy are just imaginary characters. In reality, we would need to use calculus to find the exact coordinates of the points where the tangent lines intersect the parabola.
To draw a diagram showing the tangent lines to the parabola y = x^2 that pass through the point (0, -25), we first need to find the equations of the tangent lines and then plot them on a graph.
Step 1: Find the slope of the tangent line
The slope of a tangent line to a curve is given by the derivative of the function. Let's differentiate y = x^2 to find the slope:
dy/dx = 2x
Step 2: Find the equation of the tangent line
Since the point (0, -25) lies on the tangent line, we can substitute these values into the equation y - y1 = m(x - x1) to find the equation of the tangent line:
y - (-25) = 2x(0 - 0)
y + 25 = 0
y = -25
So, the equation of the tangent line passing through (0, -25) is y = -25.
Step 3: Find the x-coordinate of the intersection points
To find the x-coordinate where the tangent line intersects the parabola, we set the equation of the tangent line equal to the equation of the parabola:
-25 = x^2
x^2 + 25 = 0
This equation does not have real solutions, so there is no intersection point between the tangent line y = -25 and the parabola y = x^2.
Step 4: Find another tangent line
To find the second tangent line, we consider the symmetry of the parabola about the y-axis. Because of symmetry, there must be another tangent line with the same slope passing through (0, -25), but on the other side of the parabola.
So, the equation of the second tangent line passing through (0, -25) is y = 25.
Step 5: Find the x-coordinate of the second intersection point
To find the x-coordinate where this second tangent line intersects the parabola, we set the equation of the tangent line equal to the equation of the parabola:
25 = x^2
x^2 - 25 = 0
(x - 5)(x + 5) = 0
So, the x-coordinates of the intersection points are x = -5 and x = 5.
The coordinates of the points where the tangent lines intersect the parabola are (-5, 25) and (5, 25).
In summary, the diagram would show a parabola y = x^2 with two tangent lines passing through the point (0, -25). The first tangent line is y = -25, and it does not intersect the parabola. The second tangent line is y = 25, and it intersects the parabola at the points (-5, 25) and (5, 25).
To answer this question, we can follow these steps:
1. Start by graphing the parabola y = x^2. Plot several points along the curve of the parabola.
2. Now we need to find the equation of the tangent lines passing through the point (0, -25).
3. Let's consider a generic point (x, y) on the parabola y = x^2. The slope of the tangent line at that point can be found using calculus or by taking the derivative of y = x^2, which gives us dy/dx = 2x.
4. Since the tangent line passes through (0, -25), we can find its slope using the formula (y - y1) / (x - x1), where (x1, y1) is the point of tangency. Hence, we have (-25 - y) / (0 - x) = 2x. Simplifying this equation gives us -25 = 2xy.
5. We can substitute y = x^2 into the equation to get -25 = 2x(x^2) = 2x^3. Rearranging the equation gives us 2x^3 + 25 = 0.
6. Now, we need to solve this equation for x to find the x-coordinates of the points of intersection. You can use various methods to solve this equation, such as synthetic division, factoring, or numerical methods (e.g., Newton's method).
7. After finding the x-coordinates of the points of intersection, substitute those values back into the equation y = x^2 to obtain the corresponding y-coordinates.
8. Plot the points of intersection on the graph of the parabola.
9. Finally, draw the tangent lines passing through (0, -25) and intersecting the parabola at the previously found points.
The resulting diagram will show the parabola, the two tangent lines passing through (0, -25), and the points where the tangent lines intersect the parabola.