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

#3**+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) / √(A^{2} + B^{2}) 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 ) / √(12^{2} + 5^{2}) = abs(169) / 13 = 13

Here's the graph ...

CPhill
Jan 26, 2015

#1**+10 **

1. Write the circle equation in the form (x-xc)^{2} + (y-yc)^{2} = r^{2}, 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} = 10^{2}, 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} ) =√( (12^{2}+5^{2}) = √169 = 13. (thanks Chris!).

.

Alan
Jan 26, 2015

#2**0 **

Very nice, Alan....!!!

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

CPhill
Jan 26, 2015

#3**+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) / √(A^{2} + B^{2}) 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 ) / √(12^{2} + 5^{2}) = abs(169) / 13 = 13

Here's the graph ...

CPhill
Jan 26, 2015