Question:

Find the equation of the parabola with focus (6, 0) and directrix x = -6. Also find the length of latus-rectum

Answer:

Answer:

1. The equation of the parabola is given by (y - 0)^2 = 4p(x - 6), where p is the distance between the focus and directrix.

2. The length of the latus-rectum is 4p.

Question:

Find the coordinates of the focus axis of the parabola, the equation of directrix and the length of the latus rectum for the parabola x^2 =−16y.

Answer:

Step 1: To find the coordinates of the focus axis of the parabola, use the formula (h, k) = (0, 4).

Step 2: To find the equation of the directrix, use the formula y = -4.

Step 3: To find the length of the latus rectum, use the formula l = 4√2.

Question:

Find the coordinates of the focus, axis of the parabola ,the equation of directrix and the length of the latus rectum for x^2=−9y

Answer:

1. The coordinates of the focus of the parabola is (0, -9/4).

2. The axis of the parabola is the x-axis.

3. The equation of the directrix is y = 9/4.

4. The length of the latus rectum is 9.

Question:

Find the equation of the parabola whose- Vertex (0,0); Focus (3,0)

Answer:

Answer: Step 1: Find the equation of the parabola in the form y = ax2 + bx + c.

Step 2: Find the value of “a” using the given information.

The vertex of the parabola is (0,0), which means that the equation of the parabola must have a c-value of 0. Therefore, the equation of the parabola can be written as y = ax2 + bx.

Step 3: Find the value of “b” using the given information.

The focus of the parabola is (3,0). This means that the equation of the parabola must have a b-value of -3. Therefore, the equation of the parabola can be written as y = ax2 - 3x.

Step 4: Find the value of “a” using the given information.

The focus of the parabola is (3,0). This means that the equation of the parabola must have an a-value of 1/4. Therefore, the equation of the parabola can be written as y = (1/4)x2 - 3x.

Therefore, the equation of the parabola whose vertex is (0,0) and focus is (3,0) is y = (1/4)x2 - 3x.

Question:

Find the equation of parabola with vertex (0,0) & focus at (0,2).

Answer:

Step 1: Identify the equation of a parabola in standard form. This is y = ax^2 + bx + c.

Step 2: Identify the vertex of the parabola. The vertex is (0,0).

Step 3: Identify the focus of the parabola. The focus is (0,2).

Step 4: Use the vertex and focus to calculate the value of the equation’s constants. The equation of the parabola is y = -(1/4)x^2 + 0x + 0.

Question:

Find the equation of the parabola that satisfies the following conditions: Vertex (0,0) passing through (2,3) and axis is along x-axis

Answer:

Answer: Step 1: Identify the vertex of the parabola. Vertex = (0,0)

Step 2: Identify the direction of the axis of the parabola. Axis = x-axis

Step 3: Identify the point that the parabola passes through. Point = (2,3)

Step 4: Substitute the values in the general equation of the parabola: y = ax^2 + bx + c

Step 5: Solve the equation for a, b, and c by substituting the values of the vertex and the point.

Substituting the vertex (0,0): 0 = a(0)^2 + b(0) + c c = 0

Substituting the point (2,3): 3 = a(2)^2 + b(2) + 0 3 = 4a + 2b

Step 6: Solve the simultaneous equations for a and b. 3 = 4a + 2b 4a + 2b = 0

Subtracting 4a from both sides: 2b = -4a b = -2a

Substituting b = -2a in the equation 3 = 4a + 2b: 3 = 4a + 2(-2a) 3 = 4a - 4a 3 = 0

a = 0.75 b = -1.5

Step 7: The equation of the parabola is: y = 0.75x^2 - 1.5x

Question:

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the rectum for y^2=12x

Answer:

1. The coordinates of the focus are (6, 0).

2. The axis of the parabola is the x-axis.

3. The equation of the directrix is x = -2.

4. The length of the rectum is 12.

Question:

Find the coordinates of the focus, axis of the parabola ,the equation of directrix and the length of the latus reactum for x^2=6y

Answer:

1. The coordinates of the focus of the parabola is (0,3).

2. The axis of the parabola is the x-axis.

3. The equation of the directrix is x = -3.

4. The length of the latus rectum is 6.

Question:

Find the coordinates of the focus axis of the parabola the equation of directrix and the length of the latus rectum for y^2=−8x

Answer:

1. The equation of the parabola is y^2=-8x.

2. The coordinates of the focus axis is (0, 4).

3. The equation of the directrix is x=-1/4.

4. The length of the latus rectum is 8.

Question:

Find the equation of the parabola that satisfies the following conditions: Vertex (0,0) passing through (5,2) and symmetric with respect to y- axis

Answer:

Step 1: Since the vertex is (0,0) and the parabola is symmetric with respect to the y-axis, the equation of the parabola should be in the form of y = ax^2.

Step 2: Substitute the given coordinates in the equation and solve for a.

y = a(x^2)

0 = a(0^2)

2 = a(5^2)

2/25 = a

Step 3: Substitute the value of a in the equation and solve for y.

y = (2/25)x^2

Therefore, the equation of the parabola is y = (2/25)x^2.

Question:

Find the coordinates of the focus axis of the parabola the equation of directrix and the length of the latus rectum for y^2=10x

Answer:

Answer: Step 1: Find the equation of the vertex: The equation of the vertex of the parabola is (0, 0).

Step 2: Find the equation of the directrix: The equation of the directrix of the parabola is x = -5.

Step 3: Find the coordinates of the focus axis: The coordinates of the focus axis of the parabola is (5, 0).

Step 4: Find the length of the latus rectum: The length of the latus rectum is 10 units.

Question:

Find the equation of the parabola that satisfies the following conditions : Focus(0,-3) and directrix y=3.

Answer:

Answer: Step 1: The equation of a parabola with focus (0,-3) and directrix y=3 is given by the formula: y^2 = 4px + 8p

Step 2: Substituting the given values in the formula, we get: y^2 = 4p(0) + 8p

Step 3: Simplifying the equation, we get: y^2 = 8p

Step 4: Thus, the equation of the parabola satisfying the given conditions is y^2 = 8p.