Find the equation of the tangent of the parabola y^2= 4px

In summary, the conversation discusses two questions involving finding the equation of a tangent to different parabolas and identifying the point of contact and the line of intersection. The first question involves finding the equation of the tangent perpendicular to a given line, while the second question involves finding the equation of the tangent at the extremities of the latus rectum. The conversation also mentions using analytic geometry to solve these problems and asks for further explanation or resources.
  • #1
callas
2
0
hello everyone,
:confused:i'm having can't seem to solve these two questions...

1) Find the equation of the tangent of the parabola y^2= 4px, perpendicular to the lines 4Y- X + 3=0, and find the point of contact?

2) Find the equation of the tangent to the parabola Y^2= 10x, at the extremities of the lactus rectum. On what line do these tangents intersect?

If there is anyone out there who knows the answer...could u please explain or direct me to a suitable site which may explain the concepts.:cry:


Thanks callas
 
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  • #2


1) Can you find the gradient of that line? It says to find the tangent parallel to that line so when you take the derivative of the parabola, you're going to have to make it equal to the gradient of the line to find the value(s) of the points on the parabola that are have this gradient.

2) Do you know how to find the equation of the latus rectum on a parabola? You'll have to again take the derivative of the parabola and then find what the gradients of the tangents are at the points where the latus rectum intersects the parabola. Since you'll know the it should be fairly simple to find the point of intersection between the tangent and the parabola, so you should be able to form an equation for each tangent. Now just find where these tangents intersect and notice the point's similarity to the focus and latus rectum.
 
  • #3


thanks...but i can not use any calculus method...it must only be solved using analytic geometry...yes i could find the gradient...
 
  • #4


callas said:
hello everyone,
:confused:i'm having can't seem to solve these two questions...

1) Find the equation of the tangent of the parabola y^2= 4px, perpendicular to the lines 4Y- X + 3=0, and find the point of contact?
Any line perpendicular to 4y- x+ 3= 0 must have slope -4 and is of the form y= -4x+ a or x= -(1/4)y+ b. That line will cross to [itex]y^2= 4px[/itex] or [itex]x= (1/4p)y^2[/itex] where [itex]x= -(1/4)y+ b= y^2/4p[/itex] and will be tangent if and only if that equation has a double root. That is equivalent to the quadratic equation [itex]y^2+ py- 4pb= 0[/itex] and has a double root if and only if the discriminant, [itex]p^2- 4(-4pb)= p^2+ 16pb= 0[/itex]. Solve that for b.

2) Find the equation of the tangent to the parabola Y^2= 10x, at the extremities of the lactus rectum. On what line do these tangents intersect?
The "latus rectum" of this parabola is the vertical line segment, passing through the focus of the parabola with endpoints on the parabola. This particular parabola has focus at (2.5, 0) and its endpoints are where [itex]y^2= 10(2.5)[/itex] or (2.5, -5) and (2.5, 5).
Any line passing through (2.5, -5) has the form x= m(y+5)+ 2.5. Again, use the fact that we must have a double root for [itex]x= (1/10)y^2= m(y+5)+ 2.5[/itex], which is equivalent to [itex]y^2- 10my+ 7= 0[/itex] to find m. Do the same with (2.5, 5).

If there is anyone out there who knows the answer...could u please explain or direct me to a suitable site which may explain the concepts.:cry:


Thanks callas
 

Related to Find the equation of the tangent of the parabola y^2= 4px

1. What is the equation of the tangent line to the parabola y^2=4px at a given point?

The equation of the tangent line to the parabola y^2=4px at a given point (x0,y0) is y = (2p)x0 + p.

2. How do you find the slope of the tangent line to the parabola y^2=4px at a given point?

The slope of the tangent line to the parabola y^2=4px at a given point (x0,y0) is equal to 2p.

3. Can you find the equation of the tangent line to the parabola y^2=4px without knowing the value of p?

No, the equation of the tangent line to the parabola y^2=4px is dependent on the value of p, which is the distance from the focus to the vertex of the parabola.

4. How can you use the equation of the tangent line to determine the coordinates of the point of tangency?

The coordinates of the point of tangency can be found by setting y = 0 in the equation of the tangent line, which will give the x-coordinate. The y-coordinate can then be found by substituting the x-coordinate into the original parabola equation.

5. Is the equation of the tangent line to the parabola y^2=4px always a straight line?

Yes, the equation of the tangent line to the parabola y^2=4px will always be a straight line as long as the parabola is not degenerate (i.e. p=0).

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