Question

Tell me some very very important properties of Parabola
10 points ​

Answer 1

Answer:

1. The graph of a quadratic function is a U-shaped curve called a parabola.

2. One important feature of the graph is that it has an extreme point, called the vertex.

3. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function.

4. In either case, the vertex is a turning point on the graph

5. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry.

6. The y-intercept is the point at which the parabola crosses the y-axis.

7. The x-intercepts are the points at which the parabola crosses the x-axis.

8. . If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of x at which y=0.

Answer 2

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✪Tangents at extremities of focal chord intersect at directrix.

✪The image of focus in any tangent lies on directrix of parabola.

✪Foot of perpendicular from focus of parabola to any tangent lies on tangent at vertex

✪Portion of tangent intercepted between directrix and point of contact substance right angle at focus of parabola

✪If n is the foot of perpendicular from focus to tangent at P then SA,SN,SP will be in G.P where S is the focus of parabola and a is the vertex of parabola

✪If T is the point of intersection of tangents at P and Q then SP,ST,SQ will be in G.P i.e ST²=SP.SQ where S is the Focus of parabola

✪Circle drawn on Focal Radii as diameter touches tangent at vertex.

✪circle drawn on focal chord and diameter always touches directrix of parabola

✪area of triangle formed by pair of tangent and chord of contact w.r.t is

$\implies \bigg| \dfrac{(y_1 ^2 -4ax_1)^\dfrac{3}{2} }{2a} \bigg| =\bigg| \dfrac{S_1}{2a} \bigg| ^\dfrac{3}{2}$

✪length of Chord of Contact with respect to (x1,y1) of parabola y²=4ax.

$\dfrac{1}{a} \sqrt{(y^2_1 -4ax_1)(y_1^2 +4a)}$