Lecture 6 [09 March, 2020] MATH 127 (Prof Farid)

 

 MATH -127 (Coordinate Geometry)

Chapter -3
The Circle



• In this figure, "O" is the pole and "OP" is the polar ( can be tangent to the circle)

• Length and midpoint of the chord intercepted by  x2+y2=a2 on the line  y=mx+c  


Midpoint  (mc1+m2,c1+m2) AB=2AC=2{a2(1+m2)c21+m2}OC y=1mx            x+my=0  



Since, the equation of AB is,  y=mx+c , the equation of the perpendicular OC is  y=1mx . Solving (2) and (3) we will get the point C. The length of OA is a (since radius). Thus we can find A.


The equation of the polar with respect to the circle is  x2+y2=a2 is  xx1+yy1=a2 at (x1,y1)  


Question 1: Find the pole of the straight line  2xy=6 with respect to the circle  5x2+5y2=9 ( ---------- )

Solution: Given equations are,

  2xy=6 5x2+5y2=9  

 Let P(x1,y1 be the pole, then the equation of the pole with respect to the circle is given by  xx1+yy1=a25xx1+5yy1=9  
    Since (1.2) & (1.3) is identical, we can write  5x12=5y11=96  
    the coordinates of the pole are  (35,310)


    ♦ Combined equation of the pair of tangents: 
  x2+y2=a2x2+y2a2=0And,x2+y2+2gx+2fy+c=0S=0 for Equation (5),SS1=T2(x2+y2a2)(x12+y12a2)=(ss1+yy1a2)2for Equation (6),SS1=T2(x2+y2+2gx+2fy+c).(x12+y12+2gx1+2fy1+c)                              ={xx1+yy1+g(x+x1)+f(y+y1)+c}  


 ---------- 



    Question 2: Find the equation of the pair of the tangent drawn from (0,0) to  x2+y2+2gx+2fy+c=0  
Solution: The equation of the pair of tangents drawn from (0,0) to  x2+y2+2gx+2fy+c=0  

  SS1=T2(x2+y2+2gx+2fy+c).c={x×0+y×0+g(x+0)+f(y+0)+c}2c.(x2+y2+2gx+2fy+c)=(gx+fy+c)2cx2+cy2+2cgx+2cfy+c2=g2x2+f2g2+c2+2gfxy+2fy+2gxccx2+cy2g2x2f2y22gxfy=0c(x2+y2)(gx+fy)2=0C.(x2+y2)=(gx+fy)2  

    Question 3: Prove that, if the polar of a point "P" with respect to the circle  x2+y2=37 touchhes the circle  (x3)2+(y+2)2=2 . The locas of "P" must be conic.

Solution: From (1) we get, 

  xx1+yy1=37|3x12y137|x12+y12(3x12y137)2=25(x12+y12)25x12+25y12+(3x1+2y1+37)2=025x12+25y12+9x12+4y12+136912x1y1+148y1111x1=034x12+29y12111x1+148y1+1369=0  





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