JEEAdvanced
PQ is a focal chord of the parabola y^2 = 8x with focus S. The length…
Question
PQ is a focal chord of the parabola y^2 = 8x with focus S. The length of the chord PQ is 32/3, and PQ is not the latus rectum. If the product SP*SQ of the two focal distances equals m/n in lowest terms, find m+n.
✓ Verified answer: 67checked by our engine — not a guess
Step-by-step solution
For y^2 = 8x we have 4a = 8, so a = 2. The semi-latus rectum is l = 2a = 4.
For any focal chord of a parabola the focal distances satisfy the harmonic relation
1/SP + 1/SQ = 2/l.
Also SP + SQ = PQ (the chord passes through the focus), so SP + SQ = 32/3.
Combine: (SP + SQ)/(SP*SQ) = 2/l => SP*SQ = (SP+SQ) * l/2 = (32/3)(4/2) = (32/3)(2) = 64/3.
Check consistency: SP and SQ are roots of z^2 - (32/3)z + 64/3 = 0, giving z = 8/3 and z = 8, both positive, so a valid non-latus-rectum chord exists.
Thus m/n = 64/3, m+n = 67.
Final answer67
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 · MathA point P moves in the plane so that the length of the tangent drawn…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 · 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