Q. 54.1( 101 Votes )

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

Answer :

Let A (4, 0, 0) & B (– 4, 0, 0)

Let the coordinates of point P be (x, y, z)

__Calculating PA__

P ≡ (x, y, z) and A ≡ (4, 0, 0)

Distance PA

Here,

x_{1} = x, y_{1} = y, z_{1} = z

x_{2} = 4, y_{2} = 0, z_{2} = 0

Distance PA

__Calculating PB__

P ≡ (x, y, z) and B ≡ (– 4, 0, 0)

Distance PB

Here,

x_{1} = x, y_{1} = y, z_{1} = z

x_{2} = – 4, y_{2} = 0, z_{2} = 0

Distance PB

Given that –

PA + PB = 10

⇒ PA = 10 – PB

Squaring both sides, we get –

PA^{2} = (10 – PB)^{2}

⇒ PA^{2} = 100 + PB^{2} – 20 PB

⇒ (4 – x)^{2} + (0 – y)^{2} + (0 – z)^{2}

= 100 + (– 4 – x)^{2} + (0 – y)^{2} + (0 – z)^{2} – 20 PB

⇒ (16 + x^{2} – 8x) + (y^{2}) + (z^{2})

= 100 + (16 + x^{2} + 8x) + (y^{2}) + (z^{2}) – 20 PB

⇒ 20 PB = 16x + 100

⇒ 5 PB = (4x + 25)

Squaring both sides again, we get –

⇒ 25 PB^{2} = 16x^{2} + 200x + 625

⇒ 25 [(– 4 – x)^{2} + (0 – y)^{2} + (0 – z)^{2}] = 16x^{2} + 200x + 625

⇒ 25 [x^{2} + y^{2} + z^{2} + 8x + 16] = 16x^{2} + 200x + 625

⇒ 25x^{2} + 25y^{2} + 25z^{2} + 200x + 400 = 16x^{2} + 200x + 625

**⇒ 9x ^{2} + 25y^{2} + 25z^{2} – 225 = 0**

**This is the required equation.**

Rate this question :

Determine the point on yz-plane which is equidistant from points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).

RD Sharma - MathematicsThe length of the perpendicular drawn from the point P(a, b, c) from z-axis is

RD Sharma - MathematicsLet (3, 4, -1) and (-1, 2, 3) be the endpoints of a diameter of a sphere. Then, the radius of the sphere is equal to

RD Sharma - MathematicsThe length of the perpendicular drawn from the point P(3, 4, 5) on the y-axis is

RD Sharma - MathematicsIf P (0, 1, 2), Q(4, -2, 1) and O(0, 0, 0) are three points, then ∠POQ =

RD Sharma - MathematicsShow that the triangle ABC with vertices A (0, 4, 1), B (2, 3, – 1) and C (4, 5, 0) is right angled.

Mathematics - ExemplarWrite the length of the perpendicular drawn from the point P(3, 5, 12) on the x-axis.

RD Sharma - MathematicsIf the distance between the points P(a, 2, 1) and Q(1, -1, 1) is 5 units, find the value of a.

RD Sharma - MathematicsFind the point on y-axis which is at a distance of units from the point (1, 2, 3).

RD Sharma - MathematicsFind the point on x-axis which is equidistant from the points A(3, 2, 2) and B(5, 5, 4).

RD Sharma - Mathematics