NIMCET 2008 Mathematics PYQ — A line has intercepts and on the coordinate axes. When the axes a… | Mathem Solvex | Mathem Solvex
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NIMCET 2008 — Mathematics PYQ
NIMCET | Mathematics | 2008
A line L has intercepts a and b on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts p and q. Which of the following statements is true?
Choose the correct answer:
A.
a2+b2=p2+q2
B.
a21+b21=p21+q21
(Correct Answer)
C.
a2+p2=b2+q2
D.
a21+p21=b21+q21
Correct Answer:
a21+b21=p21+q21
Explanation
1. Equation of the Line in Intercept Form
In the original coordinate system, the equation of the line with intercepts a and b is:
ax+by=1
This can be rewritten in general form as:
a1x+b1y−1=0
2. Distance from the Origin
The perpendicular distance (d) from the origin (0,0) to a line Ax+By+C=0 is given by:
d=A2+B2∣C∣
For our line, the distance d is:
d=(a1)2+(b1)2∣−1∣=a21+b211
Squaring both sides gives:
d21=a21+b21
3. Rotating the Axes
When the coordinate axes are rotated about the origin, the relative position of the origin and the line L does not change. Therefore, the perpendicular distance from the origin to the line remains invariant (constant).
In the new coordinate system, the intercepts are p and q. The equation of the line becomes:
px′+qy′=1
The perpendicular distance d from the origin in this system is:
d=p21+q211⟹d21=p21+q21
4. Equating the Distances
Since the distance d is the same in both coordinate systems:
a21+b21=p21+q21
Conclusion
The relationship between the intercepts before and after rotation is based on the constant perpendicular distance from the origin.
Correct Option: (b)
Explanation
1. Equation of the Line in Intercept Form
In the original coordinate system, the equation of the line with intercepts a and b is:
ax+by=1
This can be rewritten in general form as:
a1x+b1y−1=0
2. Distance from the Origin
The perpendicular distance (d) from the origin (0,0) to a line Ax+By+C=0 is given by:
d=A2+B2∣C∣
For our line, the distance d is:
d=(a1)2+(b1)2∣−1∣=a21+b211
Squaring both sides gives:
d21=a21+b21
3. Rotating the Axes
When the coordinate axes are rotated about the origin, the relative position of the origin and the line L does not change. Therefore, the perpendicular distance from the origin to the line remains invariant (constant).
In the new coordinate system, the intercepts are p and q. The equation of the line becomes:
px′+qy′=1
The perpendicular distance d from the origin in this system is:
d=p21+q211⟹d21=p21+q21
4. Equating the Distances
Since the distance d is the same in both coordinate systems:
a21+b21=p21+q21
Conclusion
The relationship between the intercepts before and after rotation is based on the constant perpendicular distance from the origin.
Correct Option: (b)
NIMCET
The equation 3x2+10xy+11y2+14x+12y+5=0 represents,
