AP EAMCET · Maths · Three Dimensional Geometry

If the equation of the plane which is at a distance of $\frac{1}{3}$ units from the origin and perpendicular to a line whose directional ratios are $(1, 2, 2)$ is $x + p y + q z + r = 0$ then $\sqrt{p^{2} + q^{2} + r^{2}} =$

A $3$
B $\sqrt{5}$
C $\sqrt{13}$
D $2$

Verified Solution

Answer & Solution

Correct Answer

(A) $3$

Step-by-step Solution

Detailed explanation

Given, Direction ratio of perpendicular line to the plane is 1,2,2, So the equation of the plane will be x+2y+2z=d Now given distance from origin is 13, So by perpendicular distance formula we get, d12+22+22=13⇒d=1 So, equation of plane will be x+2y+2z-1=0 Now comparing…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If the equation of the plane which is at a distance of 13 units from the origin and perpendicular to a line whose directional ratios are 1,2,2 is x+py+qz+r=0 then p2+q2+r2=

Step-by-step Solution

If the equation of the plane which is at a distance of $\frac{1}{3}$ units from the origin and perpendicular to a line whose directional ratios are $(1, 2, 2)$ is $x + p y + q z + r = 0$ then $\sqrt{p^{2} + q^{2} + r^{2}} =$