+0  
 
0
247
3
avatar

GIVEN THAT THE EQUATION OF CIRCLE C IS $${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{y}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{10}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{22}}{\mathtt{\,\times\,}}{\mathtt{y}}{\mathtt{\,\small\textbf+\,}}{\mathtt{46}}$$=0 

Denote the centre of C by Q

The equation of a straight line L is 12x+5y+54=0 , where L and C does not intersect.Let P be a point of L which is the closest to Q .

Find the length of PQ

Guest Jan 26, 2015

Best Answer 

 #3
avatar+80956 
+10

Here's another approach....find the circle's center as Alan did  = (5,11)

And the distance from a line to a point not on that line is given by

d = abs ( Ax + By + C) / √(A2 + B2)       where (x,y) is the given point and Ax + By + C = 0 is the equation of the line.....so we have....

d = abs (12(5) + 5(11) + 54 ) / √(122 + 52) = abs(169) / 13 = 13

Here's the graph ...

GRAPH

 

CPhill  Jan 26, 2015
Sort: 

3+0 Answers

 #1
avatar+26399 
+10

1.  Write the circle equation in the form (x-xc)2 + (y-yc)2 = r2,  where (xc, yc) are the coordinates of the centre and r is the radius.  

2. Your circle equation can be written as (x-5)2 + (y-11)2 = 102, so the centre is at (5, 11).

 

3. Your straight line can be written in the form y1 = -(12/5)x - 54/5, so it has slope -12/5.

4. The slope of the straight line perpendicular to this is y2 = (5/12)x + k, where k is a constant.

5. For this second line to go through (5, 11) we must have 11 = (5/12)*5 + k so that k = 107/12

6. The equation of the second line is therefore y2 = (5/12)x + 107/12

 

7. This hits the first line when y1=y2.  The value of x at which this occurs is given by equating the two straight line equations:  -(12/5)x - 54/5 = (5/12)x + 107/12

8.  Rearrange as;  (5/12 + 12/5)x = -54/5 - 107/12  or (169/60)x = -1183/60 so that x = -1183/169 = -7

 

9. When x = -7 then y1 = y2 = -(5/12)*7 +107/12 = 72/12 = 6

 

10. The coordinates of P are therefore (-7, 6)

 

11. The distance between P and Q is given by √( (5 - (-7))2 + (11 - 6)2 ) = 25√2 ≈ 35.355

.

Oops!  Ignore my result for 11.  It should be √( (5 - (-7))2 + (11 - 6)2 ) =√( (122+52) = √169 = 13. (thanks Chris!).

.

Alan  Jan 26, 2015
 #2
avatar+80956 
0

Very nice, Alan....!!!

I really like that one....a definite "Daily Wrap" candidate

 

CPhill  Jan 26, 2015
 #3
avatar+80956 
+10
Best Answer

Here's another approach....find the circle's center as Alan did  = (5,11)

And the distance from a line to a point not on that line is given by

d = abs ( Ax + By + C) / √(A2 + B2)       where (x,y) is the given point and Ax + By + C = 0 is the equation of the line.....so we have....

d = abs (12(5) + 5(11) + 54 ) / √(122 + 52) = abs(169) / 13 = 13

Here's the graph ...

GRAPH

 

CPhill  Jan 26, 2015

25 Online Users

avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details