If A and B be the points \[\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)\]and \[\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)\], respectively, find the equation of the set of points P such that \[\mathbf{P}{{\mathbf{A}}^{\mathbf{2}}}~+\text{ }\mathbf{P}{{\mathbf{B}}^{\mathbf{2}}}~=\text{ }{{\mathbf{k}}^{\mathbf{2}}}\], where k is a constant.

If A and B be the points

$\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)$

and

$\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)$

, respectively, find the equation of the set of points P such that

$\mathbf{P}{{\mathbf{A}}^{\mathbf{2}}}~+\text{ }\mathbf{P}{{\mathbf{B}}^{\mathbf{2}}}~=\text{ }{{\mathbf{k}}^{\mathbf{2}}}$

, where k is a constant.

Class 11 | Maths | Miscellaneous Exercise | NCERT | Three Dimensional Geometry

If A and B be the points

$\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)$

and

$\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)$

, respectively, find the equation of the set of points P such that

$\mathbf{P}{{\mathbf{A}}^{\mathbf{2}}}~+\text{ }\mathbf{P}{{\mathbf{B}}^{\mathbf{2}}}~=\text{ }{{\mathbf{k}}^{\mathbf{2}}}$

, where k is a constant.

Given:

The points A

$\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)$

and B

$\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)$

$\mathbf{P}{{\mathbf{A}}^{\mathbf{2}}}~+\text{ }\mathbf{P}{{\mathbf{B}}^{\mathbf{2}}}~=\text{ }{{\mathbf{k}}^{\mathbf{2}}}$

……….(1)

Let the point be P (x, y, z).

Now by using distance formula,

We know that the distance between two points

$P\text{ }({{x}_{1}},\text{ }{{y}_{1}},\text{ }{{z}_{1}})$

and

$Q\text{ }({{x}_{2}},\text{ }{{y}_{2}},\text{ }{{z}_{2}})$

is given by

So,

And

Now, substituting these values in (1), we have

$[{{\left( 3\text{ }\text{ }x \right)}^{2}}~+\text{ }{{\left( 4\text{ }\text{ }y \right)}^{2}}~+\text{ }{{\left( 5\text{ }\text{ }z \right)}^{2}}\left] \text{ }+\text{ } \right[{{\left( -1\text{ }\text{ }x \right)}^{2}}~+\text{ }{{\left( 3\text{ }\text{ }y \right)}^{2}}~+\text{ }{{\left( -7\text{ }\text{ }z \right)}^{2}}]\text{ }=\text{ }{{k}^{2}}$

$[(9\text{ }+\text{ }{{x}^{2}}~\text{ }6x)\text{ }+\text{ }(16\text{ }+\text{ }{{y}^{2}}~\text{ }8y)\text{ }+\text{ }(25\text{ }+\text{ }{{z}^{2}}~\text{ }10z)\left] \text{ }+\text{ } \right[(1\text{ }+\text{ }{{x}^{2}}~+\text{ }2x)\text{ }+\text{ }(9\text{ }+\text{ }{{y}^{2}}~\text{ }6y)\text{ }+\text{ }(49\text{ }+\text{ }{{z}^{2}}~+\text{ }14z)]\text{ }=\text{ }{{k}^{2}}$

$9\text{ }+\text{ }{{x}^{2}}~\text{ }6x\text{ }+\text{ }16\text{ }+\text{ }{{y}^{2}}~\text{ }8y\text{ }+\text{ }25\text{ }+\text{ }{{z}^{2}}~\text{ }10z\text{ }+\text{ }1\text{ }+\text{ }{{x}^{2}}~+\text{ }2x\text{ }+\text{ }9\text{ }+\text{ }{{y}^{2}}~\text{ }6y\text{ }+\text{ }49\text{ }+\text{ }{{z}^{2}}~+\text{ }14z\text{ }=\text{ }{{k}^{2}}$

$2{{x}^{2}}~+\text{ }2{{y}^{2}}~+\text{ }2{{z}^{2}}~\text{ }4x\text{ }\text{ }14y\text{ }+\text{ }4z\text{ }+\text{ }109\text{ }=\text{ }{{k}^{2}}$

$2{{x}^{2}}~+\text{ }2{{y}^{2}}~+\text{ }2{{z}^{2}}~\text{ }4x\text{ }\text{ }14y\text{ }+\text{ }4z\text{ }=\text{ }{{k}^{2}}~\text{ }109$

$2\text{ }({{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }{{z}^{2}}~\text{ }2x\text{ }\text{ }7y\text{ }+\text{ }2z)\text{ }=\text{ }{{k}^{2}}~\text{ }109$

$({{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }{{z}^{2}}~\text{ }2x\text{ }\text{ }7y\text{ }+\text{ }2z)\text{ }=\text{ }({{k}^{2}}~\text{ }109)/2$

Hence, the required equation is

$({{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }{{z}^{2}}~\text{ }2x\text{ }\text{ }7y\text{ }+\text{ }2z)\text{ }=\text{ }({{k}^{2}}~\text{ }109)/2$

Find the vector equation of the line passing through the point with position vector and parallel to the vector Deduce the Cartesian equations of the line.

A line passes through the point (2, 1, -3) and is parallel to the vector Find the equations of the line in vector and Cartesian forms.

A line passes through the point (3, 4, 5) and is parallel to the vector Find the equations of the line in the vector as well as Cartesian forms.

Evaluate:

If $f(x) = |x| - 3,find\mathop {\lim }\limits_{x \to 3} f(x)$

Let $\begin{array}{l} f(x) = \left\{ \begin{array}{l} \frac{x}{{|x|}},x \ne 0\\ 0,x = 0 \end{array} \right.\\ \end{array}$ . Show that $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} f(x) \end{array}$ does not exist.

Let $\begin{array}{l} f(x) = \left\{ \begin{array}{l} \frac{|x-3|}{{(x-3)'}},x \ne 3\\ 0,x = 3 \end{array} \right.\\ \end{array}$ . Show that $\begin{array}{l} \mathop {\lim }\limits_{x \to 3} f(x) \end{array}$ does not exist.

← Previous Next →

More Material

Find the angle between each of the following pairs of lines:

Find the coordinates of the foot of the perpendicular drawn from the point (1, 2, 3) to the line

Show that the lines and do not intersect each other.

Find the vector equation of the line passing through the point with position vector and parallel to the vector Deduce the Cartesian equations of the line.

A line passes through the point (2, 1, -3) and is parallel to the vector Find the equations of the line in vector and Cartesian forms.

A line passes through the point (3, 4, 5) and is parallel to the vector Find the equations of the line in the vector as well as Cartesian forms.

Evaluate:

Evaluate:

Evaluate:

If $f(x) = |x| - 3,find\mathop {\lim }\limits_{x \to 3} f(x)$

Let $\begin{array}{l} f(x) = \left\{ \begin{array}{l} \frac{x}{{|x|}},x \ne 0\\ 0,x = 0 \end{array} \right.\\ \end{array}$ . Show that $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} f(x) \end{array}$ does not exist.

Let $\begin{array}{l} f(x) = \left\{ \begin{array}{l} \frac{|x-3|}{{(x-3)'}},x \ne 3\\ 0,x = 3 \end{array} \right.\\ \end{array}$ . Show that $\begin{array}{l} \mathop {\lim }\limits_{x \to 3} f(x) \end{array}$ does not exist.

Sign up now and study all subjects for free

Class

Syllabus

Question Papers

Social Media

Apps