Prove that image of focus at any tangent to parabola lies on its directrix.
To prove that the image of the focus at any tangent to a parabola lies on its directrix, we can follow these steps:
Step 1: Understand the properties of a parabola
- The standard equation of a parabola in Cartesian coordinates is given as y^2 = 4ax, where 'a' is the distance between the vertex and the focus.
- The focus of a parabola is located at the point (a, 0), where 'a' is the distance mentioned above.
- The directrix of a parabola is a line located at a distance 'a' on the opposite side of the vertex. It is perpendicular to the axis of symmetry of the parabola.
- The tangent to a parabola is a line that touches the parabola at exactly one point.
Step 2: Find the equation of the tangent line
- Let's consider a general point on the parabola with coordinates (x, y).
- Since the point lies on the parabola, we can substitute the values into the equation of the parabola to get y^2 = 4ax.
- Differentiating both sides of the equation with respect to x will give dy/dx = 4a/x.
Step 3: Determine the equation of the tangent line
- Since the slope of the tangent line is given by dy/dx, and the point (x, y) lies on the tangent line, we can use the point-slope form of a line to find the equation of the tangent line.
- The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line.
- Substituting the values, we have y - y = (4a/x)(x - x) = 0.
- Simplifying, the equation of the tangent line becomes y - y = 0, which gives y = y.
Step 4: Find the coordinates of the image of the focus at the tangent
- Since the tangent line touches the parabola at one point, it has the same y-coordinate as the point on the parabola.
- Let's denote the x-coordinate of the point of tangency as x0.
- Since the y-coordinate of the point of tangency is the same as the y-coordinate on the parabola, we have y0^2 = 4ax0.
Step 5: Verify that the image lies on the directrix
- The image of the focus at the tangent is the point on the directrix where the tangent line intersects it.
- The equation of the directrix is given by x = -a.
- Plugging in the values, we have y = 0^2 = 4ax0 = -a.
- Simplifying, it gives x0 = -a/a = -1.
Conclusion:
- We have shown that the image of the focus at any tangent to the parabola lies on its directrix.
- By evaluating the y-coordinate of the image, we obtained y0^2 = 4ax0, which satisfies the equation of the parabola.
- Similarly, by finding the x-coordinate of the image, we obtained x0 = -1, which confirms that the image lies on the directrix with equation x = -a.