Question 70 of 70advanced🔍 AnalyzeLong Answer4 marks

An equilateral triangle is inscribed in the parabola y2=4axy^2 = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

Correct Answer

The length of the side of the equilateral triangle is 8a38a\sqrt{3}.

Exercise: Miscellaneous Exercise on Chapter 10 | Q: 8 | (Page 29)
For More Understanding

Explanation

This is a standard problem from the Parabola chapter. Since one vertex of the equilateral triangle is at the vertex of the parabola (origin), and the parabola y2=4axy^2 = 4ax is symmetric about the x-axis, the other two vertices must be symmetric points on the parabola.

Using the distance formula and the property of equilateral triangles, we can find the side length.

Solution Steps

  1. Step 1: Let the vertex of the parabola be A(0,0)A(0, 0). Since the parabola y2=4axy^2 = 4ax is symmetric about the x-axis, let the other two vertices of the equilateral triangle be B(x,y)B(x, y) and C(x,−y)C(x, -y), both lying on the parabola.

  2. Step 2: Since points BB and CC lie on the parabola y2=4axy^2 = 4ax, we have:

    y2=4axy^2 = 4ax

    which gives:

    x=y24ax = \frac{y^2}{4a}

  3. Step 3: For an equilateral triangle, all sides are equal. So AB=BCAB = BC.

  4. Step 4: Calculate:

    AB=x2+y2AB = \sqrt{x^2 + y^2}

    and

    BC=(x−x)2+(y−(−y))2=4y2=2∣y∣BC = \sqrt{(x - x)^2 + (y - (-y))^2} = \sqrt{4y^2} = 2|y|

  5. Step 5: Setting AB=BCAB = BC:

    x2+y2=2∣y∣\sqrt{x^2 + y^2} = 2|y|

    Squaring both sides:

    x2+y2=4y2x^2 + y^2 = 4y^2

    giving x2=3y2x^2 = 3y^2, so:

    x=3 ∣y∣x = \sqrt{3}\,|y|

  6. Step 6: Substituting x=y24ax = \frac{y^2}{4a}:

    3 ∣y∣=y24a\sqrt{3}\,|y| = \frac{y^2}{4a}

    giving:

    ∣y∣=4a3|y| = 4a\sqrt{3}

  7. Step 7: Therefore, x=12ax = 12a and the side length:

    AB=x2+y2=144a2+48a2=192a2=8a3AB = \sqrt{x^2 + y^2} = \sqrt{144a^2 + 48a^2} = \sqrt{192a^2} = 8a\sqrt{3}

    ∴ The length of the side of the equilateral triangle is 8a38a\sqrt{3}.

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