Answer to Question #93617 in Analytic Geometry for Kanha Sharma

Question #93617
P is a point on the plane lx + my + nz = p and a point Q is taken on the line such that OP. OQ= p^2 ; show that the locus of point Q is p(lx + my + nz)= x^2 + y^2 + z^2.
1
Expert's answer
2019-09-02T09:19:23-0400

Let P point be (α,β,γ)(\alpha,\beta,\gamma) and Q point be(x1,y1,z1)(x_1,y_1,z_1)

direction ratios of OP are α,\alpha, β,\beta, γ\gamma and direction ratios of OQ are x1,y1,z1.x_1,y_1,z_1.

Since O,Q,P are collinear, we have

αx1=βy1=γz1=k\frac{\alpha}{x_1}=\frac{\beta}{y_1}=\frac{\gamma}{z_1}=k ......(1)......(1)


As P(α,β,γ)(\alpha,\beta,\gamma) lies on the plane lx+my+nz=p,lx+my+nz=p,

So,

lα+mβ+nγ=pl\alpha+m\beta+n\gamma=p


or k(x1+my1+nz1)=pk(x_1+my_1+nz_1)=p ........(2)........(2)


Given, OP×\times OQ =p2=p^2

\therefore α2+β2+γ2(x1)2+(y1)2+(z12)=p2\sqrt{\alpha^2+\beta^2+\gamma^2}\sqrt{(x_1)^2+(y_1)^2+(z_1^2)}=p^2


or, k2(x12+y12+z12)(x1)2+(y1)2+(z12)=p2\sqrt{k^2(x_1^2+y_1^2+z_1^2)}\sqrt{(x_1)^2+(y_1)^2+(z_1^2)}=p^2


or, k(x12+y12+z12)=p2k(x_1^2+y_1^2+z_1^2)=p^2 ......(3)......(3)


On dividing (2)(2) by(3)(3) ,we get


lx1+my1+nz1x12+y12+z12=1p\frac{lx_1+my_1+nz_1}{x_1^2+y_1^2+z_1^2}=\frac{1}{p}


or, p(lx1+my1+nz1)=x12+y12+z12p(lx_1+my_1+nz_1)=x_1^2+y_1^2+z_1^2


Hence the locus of point Q is p(lx+my+nz)=x2+y2+z2.p(lx+my+nz)=x^2+y^2+z^2.


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Comments

Assignment Expert
02.09.19, 16:55

Dear Kanha Sharma, You are welcome. We are glad to be helpful. If you liked our service, please press a like-button beside the answer field. Thank you!

Kanha Sharma
02.09.19, 16:36

Thanks a lot for the solution.

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