Just Needed Help on this:
A line is Tangent to a parabola at a point P(not the Vertex) and intersects the axis of this parabola at a point Q. Prove that the line segment PQ is bisected by the line which is tangent to the parabola at its vertex.
Just Needed Help on this:
A line is Tangent to a parabola at a point P(not the Vertex) and intersects the axis of this parabola at a point Q. Prove that the line segment PQ is bisected by the line which is tangent to the parabola at its vertex.
Hello, Rimas!
A line is tangent to a parabola at a point (not the vertex)
and intersects the axis of this parabola at a point
Prove that the line segment is bisected by the line
which is tangent to the parabola at its vertex.
If they made a sketch, they could have stated the problem more clearly.The parabola is "horizontal", vertex at (0,0), opens to the right.Code:| | P o * | o * | o* (p,q) o| * o |* o | -o- - - - - - * - - - - - - - - - Q | |* | * | * |
The tangent at has its x-intercept at
Prove that is bisected by the y-axis.
We have: .
At , the slope of the tangent is: .
The equation of the tangent is: .
Since , we have: .
Let , the x-intercept is: .
Let , the y-intercept is: . .[1]
Since , then
. . Substitute into [1]: .
And is indeed the midpoint of and