Find the locus of P if P A^2+P B^2=2 k^2, where A and B are the p… - Vidyadip

Question

Find the locus of $P$ if $P A^2+P B^2=2 k^2$, where $A$ and $B$ are the points $(3,4,5)$ and $(-1,3,-7)$.

✓

Answer

Let $\mathrm{P}(\mathrm{x} 1, \mathrm{y} 1, \mathrm{z})$,
Here, $\mathrm{A}(3,4,5), \mathrm{B}(-1,3,-7) \mathrm{PA}^2+\mathrm{PB}^2=2 \mathrm{k}^2$
$\Rightarrow(\mathrm{x}-3)^2+(\mathrm{y}-4)^2+(\mathrm{z}-5)^2+(\mathrm{x}+1)^2+(\mathrm{y}-3)^2+(2+7)^2$
$=2 k^2$
$\Rightarrow x^2+9-6 x+y^2+16-8 y+z^2+25-10 z+x^2+1+2 x+y^2+9-6 y+z^2+49+14 z=2 k^2$
$\Rightarrow 2 x^2+2 y^2+2 z^2-4 x-14 y+4 z+109=2 k^2$
$\Rightarrow 2\left(x^2+y^2+z^2\right)-4 x-14 y+4 z+109-2 k^2=0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from Introduction to 3D coordinate geometry→Introduction to 3D coordinate geometry — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas: $y^2 = 8x + 8y$.

Show thet: $3(\sin\text{x}-\cos\text{x})+6(\sin\text{x}+\cos)^2+4(\sin^6\text{x}+\cos^6\text{x})=13$

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\frac{\text{px}^2\text{+qx+r}}{\text{ax+b}}$

Differentiate the following from first principle$\cos\sqrt{\text{x}}$

Express the following complex numbers in the form $\text{r}(\cos\theta+\text{i}\sin\theta):$ $1+\text{i}\tan\alpha$

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{4}}\frac{\text{x}^{3}-64}{\text{x}^2-16}$

Find the sum of the series whose $n^{th}$ term is: $2n^3 + 3n^2 - 1$

The sides of a square are $x = 6, x = 9, y = 3$ and $y = 6$. Find the equation of a circle drawn on the diagonal of the square as its diameter.

Differentiate the following functions with respect to x:$\frac{\text{4x}+5\sin\text{x}}{\text{3x}+7\cos\text{x}}$

Find the equation of the hyperbola whose,Focus is (2, 2) directrix is $\text{x+y}=\text{9}$ and eccentricity = 2