JEEAdvanced
A point P moves in the plane so that the length of the tangent drawn…
Question
A point P moves in the plane so that the length of the tangent drawn from P to the circle x^2 + y^2 = 25 is exactly twice the length of the tangent drawn from P to the circle (x-10)^2 + (y-5)^2 = 4. The locus of P is a circle. If the square of the radius of this locus circle equals m/n in lowest terms, find m+n.
✓ Verified answer: 482checked by our engine — not a guess
Step-by-step solution
The squared tangent length from P(x,y) to a circle S=0 equals S(P). So:
- to circle 1: t1 = x^2+y^2-25
- to circle 2: t2 = (x-10)^2+(y-5)^2-4
The condition (tangent to C1) = 2*(tangent to C2) gives, on squaring, t1 = 4 t2:
x^2+y^2-25 = 4[(x-10)^2+(y-5)^2-4].
Expand the right side: 4[x^2-20x+100 + y^2-10y+25 -4] = 4x^2+4y^2 -80x -40y +484.
So x^2+y^2-25 - (4x^2+4y^2-80x-40y+484) = 0 => -3x^2 -3y^2 +80x +40y -509 = 0.
Divide by -3: x^2 + y^2 - (80/3)x - (40/3)y + 509/3 = 0.
Centre = (40/3, 20/3). Radius^2 = (40/3)^2 + (20/3)^2 - 509/3 = 1600/9 + 400/9 - 1527/9 = 473/9.
So m/n = 473/9 (already lowest terms), and m+n = 473+9 = 482.
Final answer482
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Coordinate Geometry solutions
A ray of light starts at the point A(2,3). It first strikes the…JEE · MathIn triangle ABC the vertices are A(0,0), B(14,0), C(5,12). Consider…JEE · MathConsider the ellipse x^2/9 + y^2/4 = 1 with foci S and S'. Let P be…JEE · MathA circle passes through the two points (1,0) and (5,0) and is tangent…JEE · MathPQ is a focal chord of the parabola y^2 = 8x with focus S. The length…JEE · MathA single straight line is a common tangent to both parabolas y^2 = 4x…JEE · MathFor the hyperbola x^2/16 - y^2/9 = 1, the tangent at a variable point…JEE · MathConsider the family of circles passing through the two intersection…JEE · Math