NDA 2026 Mathematics PYQ — A line passes through the point and is perpendicular to the plane… | Mathem Solvex | Mathem Solvex
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NDA 2026 — Mathematics PYQ
NDA | Mathematics | 2026
A line L passes through the point (−1,2,3) and is perpendicular to the plane P given by 2x+3y+z+5=0.
What are the direction ratios of a line M parallel to the plane P?
बिंदु(−1,2,3)से गुजरने वाली एक रेखा L, समतल P पर लंब है। समतल P का समीकरण 2x+3y+z+5=0 है।
समतल P के समांतर रेखा M के दिक्-अनुपात क्या है ?
Choose the correct answer:
A.
⟨−3,2,1⟩
B.
⟨3,2,−6⟩
C.
⟨1,3,2⟩
D.
⟨2,2,−10⟩
(Correct Answer)
Correct Answer:
⟨2,2,−10⟩
Explanation
Step 1: Identify the normal vector to the plane
The equation of the plane is 2x+3y+1z+5=0.
The normal vector n to this plane is given by the coefficients of x,y, and z:
n=2i^+3j^+1k^
The direction ratios of the normal to the plane are ⟨2,3,1⟩.
Step 2: Understand the condition for parallelism
A line with direction ratios ⟨a,b,c⟩ is parallel to a plane with normal vector n=⟨nx,ny,nz⟩ if the line is perpendicular to the normal. Mathematically, the dot product must be zero:
a(nx)+b(ny)+c(nz)=0
For our plane, this condition is:
2a+3b+1c=0
Step 3: Test the given options
We check which option satisfies the equation 2a+3b+c=0:
(a) ⟨−3,2,1⟩: 2(−3)+3(2)+1(1)=−6+6+1=1=0
(b) ⟨3,2,−6⟩: 2(3)+3(2)+1(−6)=6+6−6=6=0
(c) ⟨1,3,2⟩: 2(1)+3(3)+1(2)=2+9+2=13=0
(d) ⟨2,2,−10⟩: 2(2)+3(2)+1(−10)=4+6−10=0
Since option (d) satisfies the condition 2a+3b+c=0, the direction ratios ⟨2,2,−10⟩ represent a line parallel to the plane P.
Correct Option: (d) ⟨2,2,−10⟩
Explanation
Step 1: Identify the normal vector to the plane
The equation of the plane is 2x+3y+1z+5=0.
The normal vector n to this plane is given by the coefficients of x,y, and z:
n=2i^+3j^+1k^
The direction ratios of the normal to the plane are ⟨2,3,1⟩.
Step 2: Understand the condition for parallelism
A line with direction ratios ⟨a,b,c⟩ is parallel to a plane with normal vector n=⟨nx,ny,nz⟩ if the line is perpendicular to the normal. Mathematically, the dot product must be zero:
a(nx)+b(ny)+c(nz)=0
For our plane, this condition is:
2a+3b+1c=0
Step 3: Test the given options
We check which option satisfies the equation 2a+3b+c=0:
(a) ⟨−3,2,1⟩: 2(−3)+3(2)+1(1)=−6+6+1=1=0
(b) ⟨3,2,−6⟩: 2(3)+3(2)+1(−6)=6+6−6=6=0
(c) ⟨1,3,2⟩: 2(1)+3(3)+1(2)=2+9+2=13=0
(d) ⟨2,2,−10⟩: 2(2)+3(2)+1(−10)=4+6−10=0
Since option (d) satisfies the condition 2a+3b+c=0, the direction ratios ⟨2,2,−10⟩ represent a line parallel to the plane P.
