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The curve C is given by the equations

x=12cos(t),y=5sin(t),πt<π

The line y=x intersects with C at the point P, as shown on the diagram.

Express the equation of the tangent to C at the point P in the form

ax+by+c=0,a,b,cN

where the greatest common divisor of a,b,c is 1.


Don't find t - instead, notice that, at P, we have

12cos(t)=5sin(t)

This allows you to find the exact values of sin(t) and cos(t) at P (you could, for example, square both sides, or draw a triangle).


When finding the gradient at P, notice that

12cos(t)=5sin(t)

at P, which allows you to find the exact value of cot(t).


Give the value of a+b+c.