Parametric equation of tangent question involving a circle

Question

Given a circle centered at the origin with radius $5$, find the parametric equation of the tangent line to the circle at the point $(3, −4)$.

The final equation is supposed to be like this: $$(3,-4) + t(x,y).$$

The problem is that I don't know what are the values of $x$ and $y$. Help, please.

score 1 · Accepted Answer · answered Mar 04 '18 at 05:39

1

Guide:

The radius is perpedicular to the tangent line.

Can you find a solution to $$3x-4y=0?$$

answered Mar 04 '18 at 05:39

Siong Thye Goh

153,832

since its perpendicular, i need to use dot product = 0 . and then find the the value of x and y? am i right? – Zahier Arbaai Mar 04 '18 at 06:37
yup, I am taking the dot product, you just need a solution fo $x$ and $y$ to determine the direction. – Siong Thye Goh Mar 04 '18 at 06:40

score 0 · Answer 2 · answered Mar 04 '18 at 05:39

0

Hint: Draw a radius from the origin to $(3, -4)$. This line must be perpendicular to the desired tangent line. So the two lines' corresponding direction vectors must have a dot product of zero.

Alternatively, recall that the slopes of perpendicular lines must have a product of $-1$.

answered Mar 04 '18 at 05:39

Adriano

41,969

do you mean dot product of the origin and perpendicular line = 0? because im still not quiet understand. – Zahier Arbaai Mar 04 '18 at 06:39
If $ac+bd=0$, then $ac= -bd$, if $b \neq 0$, then we have $\frac{a}{b}=-\frac{d}{c}$ or equivalently, $\frac{a}{b}\frac{c}{d}=-1$. – Siong Thye Goh Mar 04 '18 at 06:48

score 0 · Answer 3 · answered Mar 04 '18 at 06:45

0

x and y are co-ordinate points of the tangent line and their values depend on the given value for parameter (t). The parametric equation of the tangent line can be written as,

x = 3(1+t), y = -4(1-t)

The point (3,-4) represents co-ordinate point on the tangent line for t=0 Similarly, other co-ordinate points on the tangent line will be obtained for giving values for t = ....-3, -2 -1 etc., and 1, 2, 3,....etc

answered Mar 04 '18 at 06:45

BEJOY

1

Welcome to MSE. Please use MathJax. – José Carlos Santos Mar 04 '18 at 07:22

score 0 · Answer 4 · answered May 08 '24 at 18:16

0

Might be simpler:

We have the line direction vector $\mathbf{\ell}=(4,3)$ which is perperdicular to the radius vector (only swapping the $x$ and $y$ then take one negative), so you can have the line from (3,-4).

$$ (x,y)=(3,-4)+t(4,3). $$

answered May 08 '24 at 18:16

MathArt

1,568

Parametric equation of tangent question involving a circle

4 Answers4