Higher Maths - Equation of a tangent to a circle

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Schoolwork123
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#1
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#1
Hi - any ideas how I might answer this textbook question?

The circle with equation x^2 + y^2 - 10x - 14y + 29 = 0 has two tangents with gradient 2. Find the points of contact of the circle and these tangents and hence find the equations of both tangents.

The answers are:
(-1, 10); (11, 4)
y = 2x - 18; y = 2x + 12
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mqb2766
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#2
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#2
You could complete the square in x and y and get the equation of the circle in the usual form, then a bit of geometry about the radii being perpendicular to the tangents etc.
Or go down the algebra route and sub the tangent into the quadratic equation and solve for the value of c (tangent intercept) for which the discrimiant is zero.
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Schoolwork123
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#3
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#3
Thanks for your response

I've found the centre as (5, 7) and the radius 3√5. Using perpendicular gradients, the gradient of the radius perpendicular to each tangent would be -1/2. I'm not sure how to go on to find the point of contact though.
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mqb2766
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#4
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#4
(Original post by Schoolwork123)
Thanks for your response

I've found the centre as (5, 7) and the radius 3√5. Using perpendicular gradients, the gradient of the radius perpendicular to each tangent would be -1/2. I'm not sure how to go on to find the point of contact though.
A quick sketch should help. You have the line segment (center and gradient) and you know how far you go along it (radius), so ...
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Schoolwork123
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#5
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#5
Thanks so much - tried that method and it worked
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