JEE 2023 Mathematics PYQ — Let the line intersect the lines and at the points and respective… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let the line 1x=26−y=5z+8 intersect the lines 4x−5=3y−7=1z+2 and 6x+3=33−y=1z−6 at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane 2x−2y+z=14 is:
Choose the correct answer:
A.
3
B.
310
C.
4
(Correct Answer)
D.
311
Correct Answer:
4
Explanation
1. Lines ko standard form mein likhein:
Pehli line (L): 1x=−2y−6=5z+8=λ
Is par koi bhi general point: P(λ,6−2λ,5λ−8)
2. Point A nikalne ke liye (L aur dusri line ka intersection):
Dusri line: 4x−5=3y−7=1z+2=μ
Point A ke coordinates dono lines ko satisfy karenge. Solve karne par humein milta hai:
A=(1,4,−3) (yahan λ=1 hai)
3. Point B nikalne ke liye (L aur teesri line ka intersection):
Teesri line: 6x+3=−3y−3=1z−6=γ
Dono equations ko equate karne par:
B=(3,0,7) (yahan λ=3 hai)
4. Mid-point (M) nikalna:
M=(21+3,24+0,2−3+7)=(2,2,2)
5. Plane 2x−2y+z−14=0 se distance:
Point (x1,y1,z1) ki plane ax+by+cz+d=0 se distance ka formula:
Distance=a2+b2+c2∣ax1+by1+cz1+d∣
Distance=22+(−2)2+12∣2(2)−2(2)+1(2)−14∣
Distance=4+4+1∣4−4+2−14∣=3∣−12∣=312=4
Explanation
1. Lines ko standard form mein likhein:
Pehli line (L): 1x=−2y−6=5z+8=λ
Is par koi bhi general point: P(λ,6−2λ,5λ−8)
2. Point A nikalne ke liye (L aur dusri line ka intersection):
Dusri line: 4x−5=3y−7=1z+2=μ
Point A ke coordinates dono lines ko satisfy karenge. Solve karne par humein milta hai:
A=(1,4,−3) (yahan λ=1 hai)
3. Point B nikalne ke liye (L aur teesri line ka intersection):
Teesri line: 6x+3=−3y−3=1z−6=γ
Dono equations ko equate karne par:
B=(3,0,7) (yahan λ=3 hai)
4. Mid-point (M) nikalna:
M=(21+3,24+0,2−3+7)=(2,2,2)
5. Plane 2x−2y+z−14=0 se distance:
Point (x1,y1,z1) ki plane ax+by+cz+d=0 se distance ka formula:
