SOLUTION: what is the equation of the line tangent to the circle x^2+y^2=25 at the point (3,4) ? I don't know how to go about this sum. Please solve and explain thanks

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The general equation of a circle with center at and radius is



Hence the circle



Must be centered at the origin.

The line tangent to the point is perpendicular to the radius segment with endpoints and

Determine an equation of the line that contains the aforementioned radius segment by using the two point form of the equation of a line:



Where and are the coordinates of the given points.

Solve the resulting equation for to put this equation into slope-intercept form. Then determine the slope of the radius segment by inspection of the coefficient on .

Next use the fact that the slopes of two perpendicular lines are negative reciprocals, that is:



Calculate the negative reciprocal of the slope of the radius segment to get the slope of the tangent.

Then use the point-slope form of the equation of a line with the calculated slope and the point to derive the desired equation:



Where is the calculated slope and is the point

John

You can put this solution on YOUR website!
You need to implicit differentiation for this problem and solve for
So we have.

The derivative is the 2y produces the because y is a function so you have to apply the chain rule to it.
So now.



And x and y are given by (3,4) giving x=3 and y=4. (,)
So
Now for the tangent line you have.

so we have
Giving us Add 4 to both sides. 4 can be rewritten as


And that is your tangent line.