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12.  Find the equation of the parabola that satisfies the given conditions:

Vertex (0,0), passing through (5,2) and symmetric with respect to y-axis.

Given a parabola, with Vertex (0,0), passing through (5,2) and symmetric with respect to the y-axis. Since the parabola is symmetric with respect to Y=axis, it's axis will ve Y-axis. and since it passes through the point (5,2), it must go through the first quadrant. So the standard equation of such parabola is  Now since this parabola is passing through (5,2) Hence the equation of the...

11. Find the equation of the parabola that satisfies the given conditions:

Vertex (0,0) passing through (2,3) and axis is along x-axis.

Given The Vertex of the parabola is  (0,0). The parabola is passing through (2,3) and axis is along the x-axis, it will open towards right. and the standard equation of such parabola is  Now since it passes through (2,3)  So the Equation of Parabola is ;

10. Find the equation of the parabola that satisfies the given conditions:

Vertex (0,0); focus (-2,0)

Given,  Vertex (0,0) And  focus (-2,0) As vertex of the parabola is (0,0) and focus lies in the negative X-axis, The parabola will open towards left, And the standard equation of such parabola is  Here it can be seen that  So, the equation of a parabola is  .

9. Find the equation of the parabola that satisfies the given conditions:

Vertex (0,0); focus (3,0)

Given,  Vertex (0,0) And  focus (3,0) As vertex of the parabola is (0,0) and focus lies in the positive X-axis, The parabola will open towards the right, And the standard equation of such parabola is  Here it can be seen that  So, the equation of a parabola is  .

8. Find the equation of the parabola that satisfies the given conditions:

Focus (0,–3); directrix $y = 3$

Given,in a parabola, Focus : Focus (0,–3); directrix  Here, Focus is of the form (0,-a), which means it lies on the Y-axis. And Directrix is of the form  which means it lies above X-Axis. These are the conditions when the standard equation of a parabola is  . Hence the Equation of Parabola is   Here, it can be seen that: Hence the Equation of the Parabola is: .

7.  Find the equation of the parabola that satisfies the given conditions:

Focus (6,0); directrix $x = - 6$

Given, in a parabola, Focus : (6,0) And Directrix :  Here, Focus is of the form (a, 0), which means it lies on the X-axis. And Directrix is of the form  which means it lies left to the Y-Axis. These are the condition when the standard equation of a parabola is. Hence the Equation of Parabola is   Here, it can be seen that: Hence the Equation of the Parabola is: .

6. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.

$x^2 = -9y$

Given, a parabola with equation  This is parabola of the form  which opens downwards. So By comparing the given parabola equation with the standard equation, we get, Hence, Coordinates of the focus : Axis of the parabola: It can be seen that the axis of this parabola is Y-Axis. The equation of the directrix The length of the latus rectum: .

5. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.

$y^2 = 10x$

Given, a parabola with equation  This is parabola of the form  which opens towards the right. So, By comparing the given parabola equation with the standard equation, we get, Hence, Coordinates of the focus : Axis of the parabola: It can be seen that the axis of this parabola is X-Axis. The equation of the directrix The length of the latus rectum: .

3.  Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.

$y^2 = -8x$

Given, a parabola with equation  This is parabola of the form  which opens towards left. So, By comparing the given parabola equation with the standard equation, we get, Hence, Coordinates of the focus : Axis of the parabola: It can be seen that the axis of this parabola is X-Axis. The equation of the directrix The length of the latus rectum: .

2. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.

$x^2 = 6y$

Given, a parabola with equation  This is parabola of the form  which opens upward. So, By comparing the given parabola equation with the standard equation, we get, Hence, Coordinates of the focus : Axis of the parabola: It can be seen that the axis of this parabola is Y-Axis. The equation of the directrix The length of the latus rectum: .

1. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.

$y^2 =12x$

Given, a parabola with equation  This is parabola of the form  which opens towards the right. So, By comparing the given parabola equation with the standard equation, we get, Hence, Coordinates of the focus : Axis of the parabola: It can be seen that the axis of this parabola is X-Axis. The equation of the directrix The length of the latus rectum: .
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