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Prove that if a plane has intercepts a,b...

Prove that if a plane has intercepts a,b,c and is at a distance of p units from the origin, then `1/a^2+1/b^2+1/c^2=1/p^2`.

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The correct Answer is:
`(1)/(p^(2))`.
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If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that (1)/(p^(2)) = (1)/(a^(2)) +(1)/(b^(2)) .

If p_1, p_2, p_3 are the altitudes of a triangle from the vertices A, B, C and triangle , the area of the triangle, prove that : 1/p_1+1/p_2-1/p_3= (2ab)/((a+b+c)triangle) cos^2 frac (C)(2) .

A variable plane which is at a constant distance 6 p from the origin meets the axes in points A, B and C respectively. Show that the locus of the centroid of Delta ABC is 1/x^2+1/y^2+1/z^2=1/(4p^2)

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If p_1, p_2, p_3 are the altitudes of a triangle from the vertices A, B, C and triangle , the area of the triangle, prove that : p_1^-2 + p_2^-2 + p_3^-2 = ((cot A+cot B+ cotC))/triangle .

OMEGA PUBLICATION-THREE DIMENSIONAL GEOMETRY -Questions
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  2. If a line has direction ratios -18,12,-4, then what are its direction ...

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  3. Find the direction cosines of the sides of the triangle whose vertices...

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  4. A line passes through the point with position vector 2hat(i)-hat(j)+4h...

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  5. Show that the line through the points : (4,7,8), (2,3,4) is parallel t...

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  6. Find the cartesian equation of the line which passes through the point...

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  7. The vector equation of the line (x-5)/(3)=(y+4)/(7)=(z-6)/(2) is ........

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  8. Find the vector and the cartesian equations of the line that passes th...

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  9. Find the angle between the pair of line vec(r)=2hat(i)-5hat(j)+hat(k...

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  11. Find the angle between the pair of lines (x+3)/3 = (y-1)/5 = (z+3)/4an...

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  12. Find the angle between the pair of lines (x-2)/2=(y-1)/5=(z+3)/(-3) an...

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  13. Find the angle between the lines (x)/(2)=(y)/(2)=(z)/(1) and (x-5)/(4)...

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  14. Find the angle between the lines (x+1)/(2)=(y)/(3)=(z-3)/(6) and the p...

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  15. The vector equation of two lines are vec(r)=hat(i)+2hat(j)+hat(k)+la...

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  16. Find the shortest distance between the lines whose vector equations ar...

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  17. Find the shortest distance between the following two lines vec(r)=(1+2...

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  18. The vector equation of two lines are vec(r)=(1-t)hat(i)+(t-2)hat(j)...

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  19. Find the equation of plane passing through (a,b,c) and parallel to the...

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  20. In the following cases, find the co-ordinates of the foot of the perpe...

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  21. In the following case, find the coordinates of the foot of the perpend...

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