Question 14 of 70intermediate🔧 ApplyShort Answer2 marks

In each of the Exercises 7 to 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.

Correct Answer

Since the vertex is at (0,0) and the parabola is symmetric with respect to the y-axis, the equation is of the form x2=4ayx^2 = 4ay or x2=−4ayx^2 = -4ay. The parabola passes through the point (5,2), which lies in the first quadrant. Since the y-coordinate is positive, the parabola opens upwards. Thus, the equation is of the form x2=4ayx^2 = 4ay.

Substituting the point (5,2) into the equation: (5)2=4a(2)(5)^2 = 4a(2) 25=8a25 = 8a a=258a = \frac{25}{8}

Therefore, the required equation of the parabola is: x2=4(258)yx^2 = 4\left(\frac{25}{8}\right)y x2=252yx^2 = \frac{25}{2}y Or, 2x2=25y2x^2 = 25y.

Exercise: EXERCISE 10.2 | Q: 12 | (Chapter: 12)
For More Understanding

Explanation

The solution follows the logic presented in Example 8 of the provided context. Because the vertex is at the origin and the axis of symmetry is the y-axis, the standard equation forms x2=4ayx^2 = 4ay or x2=−4ayx^2 = -4ay are used. The sign is determined by the location of the point (5,2); since the point has a positive y-coordinate, the parabola must open upwards, leading to the form x2=4ayx^2 = 4ay. Solving for 'a' yields the final equation.

Solution Steps

  1. Step 1: Identify the standard form. Since the vertex is (0,0) and the axis is the y-axis, the equation is x2=4ayx^2 = 4ay or x2=−4ayx^2 = -4ay.

  2. Step 2: Determine the direction. The point (5,2) has a positive y-coordinate, so the parabola opens upwards. The equation is x2=4ayx^2 = 4ay.

  3. Step 3: Substitute the coordinates (5,2) into x2=4ayx^2 = 4ay: 52=4a(2)5^2 = 4a(2).

  4. Step 4: Solve for 'a': 25=8a25 = 8a, so a=25/8a = 25/8.

  5. Step 5: Substitute 'a' back into the equation: x2=4(25/8)yx^2 = 4(25/8)y, which simplifies to 2x2=25y2x^2 = 25y.

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