JEE 2023 Mathematics PYQ — Let the tangents at the points and on the circle , intersect at t… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let the tangents at the points A(4,−11) and B(8,−5) on the circle x2+y2−3x+10y−15=0, intersect at the point C. Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to
Choose the correct answer:
A.
213
B.
13
C.
433
D.
3213
(Correct Answer)
Correct Answer:
3213
Explanation
1. Tangents ki equations nikalna
Circle ki equation hai: x2+y2−3x+10y−15=0.
Kisi point (x1,y1) par tangent ki equation hoti hai:
xx1+yy1−23(x+x1)+5(y+y1)−15=0
Point A(4,−11) par tangent (T1):
4x−11y−23(x+4)+5(y−11)−15=0
Simplify karne par: 5x−12y−152=0 --- (Eq. 1)
Point B(8,−5) par tangent (T2):
8x−5y−23(x+8)+5(y−5)−15=0
Simplify karne par: 13x−104=0⟹x=8 --- (Eq. 2)
2. Intersection Point C nikalna
x=8 ko (Eq. 1) mein rakhen:
5(8)−12y−152=0
40−152=12y⟹−112=12y⟹y=−328
Toh point C ke coordinates hain (8,−328).
3. Line AB ki equation nikalna
Points A(4,−11) aur B(8,−5) ko jodne waali line:
Slope m=8−4−5−(−11)=46=23
Equation: y+5=23(x−8)
2y+10=3x−24⟹3x−2y−34=0
4. Radius nikalna
Humein us circle ka radius chahiye jiska center C hai aur line AB uski tangent hai. Yeh radius point C se line AB ki perpendicular distance ke barabar hoga.
Kisi point (x1,y1) par tangent ki equation hoti hai:
xx1+yy1−23(x+x1)+5(y+y1)−15=0
Point A(4,−11) par tangent (T1):
4x−11y−23(x+4)+5(y−11)−15=0
Simplify karne par: 5x−12y−152=0 --- (Eq. 1)
Point B(8,−5) par tangent (T2):
8x−5y−23(x+8)+5(y−5)−15=0
Simplify karne par: 13x−104=0⟹x=8 --- (Eq. 2)
2. Intersection Point C nikalna
x=8 ko (Eq. 1) mein rakhen:
5(8)−12y−152=0
40−152=12y⟹−112=12y⟹y=−328
Toh point C ke coordinates hain (8,−328).
3. Line AB ki equation nikalna
Points A(4,−11) aur B(8,−5) ko jodne waali line:
Slope m=8−4−5−(−11)=46=23
Equation: y+5=23(x−8)
2y+10=3x−24⟹3x−2y−34=0
4. Radius nikalna
Humein us circle ka radius chahiye jiska center C hai aur line AB uski tangent hai. Yeh radius point C se line AB ki perpendicular distance ke barabar hoga.
