NIMCET 2018 Mathematics PYQ — The circles whose equations are and , will touch one another exte… | Mathem Solvex | Mathem Solvex
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NIMCET 2018 — Mathematics PYQ
NIMCET | Mathematics | 2018
The circles whose equations are x² + y² + c² = 2ax and x² + y² + c² = 2by, will touch one another externally if:
Choose the correct answer:
A.
b21+c21=a21
B.
c21+a21=b21
C.
a21+b21=c21
(Correct Answer)
D.
None of these.
Correct Answer:
a21+b21=c21
Explanation
Concept: - The equation of a circle with center at O(a, b) and radius r, is given by: (x - a)² + (y - b)² = r². - If two circles touch each other externally, then the distance between their centers is equal to the sum of their radii. - Distance Formula: The distance 'd' between two points (x₁, y₁) and (x₂, y₂) is obtained by using the Pythagoras' Theorem: d2=(x1−x2)2+(y1−y2)2
Calculation:
The equation of the first circle is: C1→x2+y2+c2=2ax
⇒x2+y2−2ax=−c2
⇒x2- 2ax+ a2+ y2= - c2+ a2
⇒(x−a)2+(y−0)2=(a2−c2)2
⇒Center is O1(a,0) and radius is r1=a2−c2.
Similarly, for the second circle C2→x2+y2+c2=2by
⇒Center is O2(0,b) and radius is r2=b2−c2.
Since the two circles touch each other externally, we must have:
Distance between O1 and O2=r1+r2
⇒a2+b2=a2−c2+b2−c2+2(a2−c2)(b2−c2)
⇒c2=(a2−c2)(b2−c2)
\text{Squaring both sides again, we get:}
⇒c4=(a2−c2)(b2−c2)
⇒c4=a2b2−a2c2−c2b2+c4
⇒a2b2=a2c2+c2b2
\text{On dividing by } a2b2c2, \text{ we get:}
⇒c21=a21+b21
Explanation
Concept: - The equation of a circle with center at O(a, b) and radius r, is given by: (x - a)² + (y - b)² = r². - If two circles touch each other externally, then the distance between their centers is equal to the sum of their radii. - Distance Formula: The distance 'd' between two points (x₁, y₁) and (x₂, y₂) is obtained by using the Pythagoras' Theorem: d2=(x1−x2)2+(y1−y2)2
Calculation:
The equation of the first circle is: C1→x2+y2+c2=2ax
⇒x2+y2−2ax=−c2
⇒x2- 2ax+ a2+ y2= - c2+ a2
⇒(x−a)2+(y−0)2=(a2−c2)2
⇒Center is O1(a,0) and radius is r1=a2−c2.
Similarly, for the second circle C2→x2+y2+c2=2by
⇒Center is O2(0,b) and radius is r2=b2−c2.
Since the two circles touch each other externally, we must have:
