Question

4y+3x =25 is the tangent to the circle x^2+ y^2 = 25 at the point P(3,4). The equation of the radius of the circle that passes through P is

PLEASE SHOW SOLUTION!

Answer 1

4x - 3y = 0

Explanation:

The angle between the radius and the tangent at P is right

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Rearrange 4y + 3x = 25 into this form

Subtract 3x from both sides

4y = - 3x + 25 ( divide all terms by 4 )

y = -
`(3)/(4)` x +
`(25)/(4)` ← in slope- intercept form

with slope m = -
`(3)/(4)`

Given a line with slope m then the slope of a line perpendicular to it is

`m_(perpendicular)` = -
`(1)/(m)` = -
`(1)/(-(3)/(4) )` =
`(4)/(3)` , thus

y =
`(4)/(3)` x + c ← is the partial equation

To find c substitute P(3, 4) into the partial equation

4 = 4 + c ⇒ c = 4 - 4 = 0

y =
`(4)/(3)` x ← equation of radius in slope- intercept form

Multiply through by 3

3y = 4x ( subtract 3y from both sides )

4x - 3y = 0 ← equation of radius in standard form