Circle C equation $(x+5)^2+(y-9)^2=25$
A line through the point P(8, -7) is a tangent to the circle C at the point T.
Find the length of PT.
The question itself is easy when using pythagoras,
But I cannot understand some of the alternative methods written in the solution paper:
"Substitutes (8, -7) into circle equation so $PT^2 = 8^2 + (−7)^2 +10×8−18×(−7) + 81$"
what is this method?
And further:
"A significant number “found” the coordinates of one point of contact T, (–8, 5), often just stating it. A number then tried to explain the solution with 3 4 5 triangles, to which the points concerned lent themselves. It was noticeable that the coordinates of the other point of contact were never found in this way."
How do I find the co-ordinate of T is (-8,5)? given that some people just stated it there must be a quick way.
Only method I can think of is horrible simultaneous equations:
differentiate the circle equation and equate that with line equation $y + 7 = \dfrac {9-T_x} {-5-T_y}(x-8) $
But I have problem finding gradient alone in this case because there are too many unknowns.
Many thanks in advance.
Best Answer
Calculate Power of the circle by shifting P to origin and the circle also accordingly for same relative position.
$$ ( x + 5 + 8)^2 + ( y -9 -7)^2 = 25 $$
Tangent length is square root of Power = $ \sqrt{13^2 + 16^2 -25} =20. $ by the Pole/Polar method.