Question 9 of 13intermediate🔧 ApplyLong Answer4 marks

Find the equation of the set of points P, the sum of whose distances from A (4,0,0)(4, 0, 0) and B (−4,0,0)(-4, 0, 0) is equal to 10.

Correct Answer

The equation of the set of points is 9x2+25y2+25z2=2259x^2 + 25y^2 + 25z^2 = 225 or equivalently x225+y29+z29=1\dfrac{x^2}{25} + \dfrac{y^2}{9} + \dfrac{z^2}{9} = 1.

Exercise: EXERCISE 11.2 | Q: 5 | (Page Page 6)
For More Understanding

Explanation

This question asks for the locus of points where the sum of distances from two fixed points A(4,0,0)A(4,0,0) and B(−4,0,0)B(-4,0,0) equals 10. Following the pattern from Example 8 in the context, we use the distance formula in three dimensions.

The key steps involve setting up the equation PA+PB=10PA + PB = 10, isolating one radical, squaring both sides, and simplifying to arrive at the standard form of an ellipsoid.

Solution Steps

  1. Step 1: Let P(x,y,z)P(x, y, z) be any point such that PA+PB=10PA + PB = 10, where A(4,0,0)A(4, 0, 0) and B(−4,0,0)B(-4, 0, 0).

  2. Step 2: Using the distance formula, PA=(x−4)2+y2+z2PA = \sqrt{(x-4)^2 + y^2 + z^2} and PB=(x+4)2+y2+z2PB = \sqrt{(x+4)^2 + y^2 + z^2}.

  3. Step 3: Given condition:

    (x−4)2+y2+z2+(x+4)2+y2+z2=10\sqrt{(x-4)^2 + y^2 + z^2} + \sqrt{(x+4)^2 + y^2 + z^2} = 10

  4. Step 4: Rearranging:

    (x−4)2+y2+z2=10−(x+4)2+y2+z2\sqrt{(x-4)^2 + y^2 + z^2} = 10 - \sqrt{(x+4)^2 + y^2 + z^2}

  5. Step 5: Squaring both sides:

    (x−4)2+y2+z2=100−20(x+4)2+y2+z2+(x+4)2+y2+z2(x-4)^2 + y^2 + z^2 = 100 - 20\sqrt{(x+4)^2 + y^2 + z^2} + (x+4)^2 + y^2 + z^2

  6. Step 6: Simplifying:

    (x−4)2−(x+4)2−100=−20(x+4)2+y2+z2(x-4)^2 - (x+4)^2 - 100 = -20\sqrt{(x+4)^2 + y^2 + z^2}

  7. Step 7: This gives:

    −16x−100=−20(x+4)2+y2+z2-16x - 100 = -20\sqrt{(x+4)^2 + y^2 + z^2}

    so

    4x+255=(x+4)2+y2+z2\frac{4x + 25}{5} = \sqrt{(x+4)^2 + y^2 + z^2}

  8. Step 8: Squaring again:

    (4x+25)225=(x+4)2+y2+z2\frac{(4x + 25)^2}{25} = (x+4)^2 + y^2 + z^2

  9. Step 9: Expanding and simplifying:

    16x225+8x+25=x2+8x+16+y2+z2\frac{16x^2}{25} + 8x + 25 = x^2 + 8x + 16 + y^2 + z^2

  10. Step 10: Final equation:

    9x2+25y2+25z2=225\mathbf{9x^2 + 25y^2 + 25z^2 = 225}

    or

    x225+y29+z29=1\mathbf{\frac{x^2}{25} + \frac{y^2}{9} + \frac{z^2}{9} = 1}
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