JEE 2023 Mathematics PYQ — Let the equations of two adjacent sides of a parallelogram be and… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let the equations of two adjacent sides of a parallelogram ABCD be 2x−3y=−23 and 5x+4y=23. If the equation of its one diagonal AC is 3x+7y=23 and the distance of A from the other diagonal is d, then 50d2 is equal to _______.
Choose the correct answer:
A.
529
(Correct Answer)
B.
592
C.
925
D.
None
Correct Answer:
529
Explanation
1. Vertices nikalna
Maan lijiye parallelogram ABCD hai. Di gayi sides:
Side 1 (L1): 2x−3y=−23
Side 2 (L2): 5x+4y=23
Diagonal (AC): 3x+7y=23
Vertex A nikalne ke liye:
Hume woh side dhundni hogi jo diagonal AC ke saath intersect karti hai. Check karte hain:
L1 aur AC ka intersection: 2x−3y=−23 aur 3x+7y=23. Solve karne par x=−4,y=5. Toh A=(−4,5).
L2 aur AC ka intersection: 5x+4y=23 aur 3x+7y=23. Solve karne par x=3,y=2. Toh C=(3,2).
Vertex B nikalne ke liye:
L1 aur L2 ka intersection point B (ya D) hoga:
2x−3y=−23 aur 5x+4y=23 ko solve karne par hume milta hai B=(−1,7).
2. Dusri Diagonal (BD) ki Equation
Parallelogram ke diagonals ek dusre ko bisect karte hain. Maan lijiye intersection point M hai.
M=Midpoint of AC=(2−4+3,25+2)=(−21,27).
Kyunki M, diagonal BD ka bhi midpoint hai, isliye line BD, points B(−1,7) aur M(−1/2,7/2) se guzregi.
Slope of BD (m):
m=−1/2−(−1)7/2−7=1/2−7/2=−7
Equation of BD:
y−7=−7(x+1)⟹y−7=−7x−7⟹7x+y=0
3. Distance d aur 50d2 ki value
Hume A(−4,5) ka distance line 7x+y=0 se nikalna hai:
d=72+12∣7(−4)+1(5)∣=49+1∣−28+5∣=5023
Ab value nikalte hain:
d2=50232=50529
50d2=50×50529=529
Uttar (Final Answer):
Iska sahi uttar 529 hai.
Explanation
1. Vertices nikalna
Maan lijiye parallelogram ABCD hai. Di gayi sides:
Side 1 (L1): 2x−3y=−23
Side 2 (L2): 5x+4y=23
Diagonal (AC): 3x+7y=23
Vertex A nikalne ke liye:
Hume woh side dhundni hogi jo diagonal AC ke saath intersect karti hai. Check karte hain:
L1 aur AC ka intersection: 2x−3y=−23 aur 3x+7y=23. Solve karne par x=−4,y=5. Toh A=(−4,5).
L2 aur AC ka intersection: 5x+4y=23 aur 3x+7y=23. Solve karne par x=3,y=2. Toh C=(3,2).
Vertex B nikalne ke liye:
L1 aur L2 ka intersection point B (ya D) hoga:
2x−3y=−23 aur 5x+4y=23 ko solve karne par hume milta hai B=(−1,7).
2. Dusri Diagonal (BD) ki Equation
Parallelogram ke diagonals ek dusre ko bisect karte hain. Maan lijiye intersection point M hai.
M=Midpoint of AC=(2−4+3,25+2)=(−21,27).
Kyunki M, diagonal BD ka bhi midpoint hai, isliye line BD, points B(−1,7) aur M(−1/2,7/2) se guzregi.
Slope of BD (m):
m=−1/2−(−1)7/2−7=1/2−7/2=−7
Equation of BD:
y−7=−7(x+1)⟹y−7=−7x−7⟹7x+y=0
3. Distance d aur 50d2 ki value
Hume A(−4,5) ka distance line 7x+y=0 se nikalna hai:
