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# circles and tangents..? help? watch

1. just wondering if anyone can help me....

1) FIND THE EQUATIONS TO THE TANGENTS TO THE CIRCLE

x^2 + y^2 = 25

FROM THE POINT P(2,11), OUTSIDE THE CIRCLE

and...

2) SHOW THAT THE STRAIGHT LINE

x - 3y - 10 = 0

IS A TANGENT TO THE CIRCLE

x^2 + y^2 = 10

BY: (I) FINDING THE DISTANCE OF THE LINE FROM THE ORIGIN
(II) SHOWING THAT THE LINE MEETS THE CIRCLE AT ONE POINT ONLY

FURTHER, DETERMINE THE EQUATION TO THE NORMAL TO THE CIRCLE THROUGH THE POINT OF CONTACT OF THE TANGENT

got really stuck...thanks for any help
2. ooh and particularly the first one...im really stuck
3. (Original post by CCDB)
just wondering if anyone can help me....

1) FIND THE EQUATIONS TO THE TANGENTS TO THE CIRCLE

x^2 + y^2 = 25

FROM THE POINT P(2,11), OUTSIDE THE CIRCLE
you should know givens you the gradients of the tangents to the curve, so using implicit differentiation (let me know if you need me to explain this concept, i'll assume you know it already to save time):

Spoiler:
Show

subbing in the values and you get

now using the formula to find an equation of a line we get

or [latex]11y + 2x - 125 = 0

4. (Original post by CCDB)
2) SHOW THAT THE STRAIGHT LINE

x - 3y - 10 = 0

IS A TANGENT TO THE CIRCLE

x^2 + y^2 = 10

BY: (I) FINDING THE DISTANCE OF THE LINE FROM THE ORIGIN
(II) SHOWING THAT THE LINE MEETS THE CIRCLE AT ONE POINT ONLY

FURTHER, DETERMINE THE EQUATION TO THE NORMAL TO THE CIRCLE THROUGH THE POINT OF CONTACT OF THE TANGENT
part i

Spoiler:
Show

the shortest distance from the line to the origin will be perpendicular to the line, therefore this line will have a gradient of -3 (product of gradients of perpendicular lines is -1) also this line must pass the origin. the equation of the line is and if we equate this to the original line we get subbing into the original line we can find . we get using pythagorus we can find the shortest distance from the origin which is this is the same as the radius of the circle

part ii

Spoiler:
Show

equate the lines to find any points of intersect.

(eq 1) and (eq 2) (rearranged)

subbing x from eq 2 into eq 1 we get

and (to get this sub into eq 2 or 1)

so theres only one point of intersect it is a tangent

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