Class 12

Math

3D Geometry

Three Dimensional Geometry

Write the equation of the plane whose intercepts on the coordinate axes are $2,−4$ and $5$ respectively.

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A plane passes through a fixed point $(a,b,c)˙$ The locus of the foot of the perpendicular to it from the origin is a sphere of radius a. $21 a_{2}+b_{2}+c_{2} $ b. $a_{2}+b_{2}+c_{2} $ c. $a_{2}+b_{2}+c_{2}$ d. $21 (a_{2}+b_{2}+c_{2})$

Find the vector equation of the following planes in Cartesian form: $r=i^−j^ +λ(i^+j^ +k^)+μ(i^−2j^ +3k^)˙$

Find the equation of a plane which passes through the point $(1,2,3)$ and which is equally inclined to the planes $x−2y+2z−3=0and8x−4y+z−7=0.$

Find the equation of the plane passing through the straight line $2x−1 =−3y+2 =5z $ and perpendicular to the plane $x−y+z+2=0.$

The foot of the perpendicular drawn from the origin to a plane is $(1,2,−3)˙$ Find the equation of the plane. or If $O$ is the origin and the coordinates of $P$ is $(1,2,−3),$ then find the equation of the plane passing through $P$ and perpendicular to $OP˙$

A point $P(x,y,z)$ is such that $3PA=2PB,$ where $AandB$ are the point $(1,3,4)and(1,−2,−1),$ irrespectivley. Find the equation to the locus of the point $P$ and verify that the locus is a sphere.

Find the plane of the intersection of $x_{2}+y_{2}+z_{2}+2x+2y+2=0and4x_{2}+4y_{2}+4z_{2}+4x+4y+4z−1=0.$

A line passes through the points $(6,−7,−1)and(2,−3,1)˙$ Find te direction cosines off the line if the line makes an acute angle with the positive direction of the x-axis.