Solve this :

Q. The common tangent to the parabola circle x2 + y2 = 4, x2 = 4y and intersect at point P, then the distance of P from origin.
Ans   :   2 2 + 1

Equation of tangent to parabola x2=4ay is given byx=ym+amy=mx-am2Herea=1Tangent, L:y-mx+m2=0If L is tangent to circle x2+y2=4, then distance of L from center of circle should be equal to radiusCenter of circle=0,0Radius=2Distance of a line ax+by+c=0 from point x1, y1=ax1+by1+ca2+b20-0+m21+m2=2m2=21+m2Squaring both sidesm4=4+4m2m4-4m2-4=0m22-4m2-4=0By quadric formulam2=4±16+162m2=4±322m2=4±422m2=2±22Since m20thereforem2=2+22m=±2+22Thus we get two tangents corresponding to each value of mL1: y-2+22x+2+22=0L2: y+2+22x+2+22=0Solving L2 and L1. Add L1 and L22y+0+22+22=0y=-2+22Put in L1-2+22-2+22x+2+22=0 -2+22x=0x=0Hence point of intersection of tangency,Q=0,-2+22Distance of Q from O=0-02+-2+22-02=-2+222=2+222=2+22=21+2

  • 0
What are you looking for?