Question

If the equation of one tangent to the circle with centre at (-2,-1) from the origin is 3x+y=0

Answer 1

Answer:

The equation of other tangent OB passing through origin is x - 3 y = 0

Step-by-step explanation:

Given as :

The center of a circle = - 2 , - 1

The equation of one tangent to the circle from origin is 3 x + y = 0

Now, As the equation of one tangle is 3 x + y = 0

Let The slop of one tangent = m

So, The tangent equation can be written y = - 3 x

And The equation of line is y = m x + c , where m is slope

So, The slope of tangent = m = - 3

Again

Let The slope of other tangent = M

So, Slope of line = M = $\dfrac{y_2-y_1}{x_2-x_1}$

Or, M = $\dfrac{0+1}{0+2}$

So Slope =M = $\dfrac{1}{2}$

So, Tan Ф = $\dfrac{m+ M}{1 - m M}$

Or,  Tan Ф = $\frac{-3 + \frac{1}{2}}{1 +3\times \frac{1}{2}}$

Or, Tan Ф = $\dfrac{-5}{5}$

So,  Ф = $Tan^{-}$ 1

Or, Ф  = - 45°

So, ∠ AOB = 2 ×Ф

Or,  ∠ AOB = 2 × (-45°)

Or,  ∠ AOB = - 90°

Thus , Both the tangents are perpendicular to each other

So, The slop of other tangent become M = $\dfrac{1}{3}$

As for perpendicular condition, product of slops of line = - 1

So, The equation of tangent with slope $\dfrac{1}{3}$ and passing through origin ( 0, 0)

y - 0 = $\dfrac{1}{3}$ × (x - 0)

Or, 3 y = x

i.e x - 3 y = 0

Hence, The equation of other tangent OB passing through origin is x - 3 y = 0 . Answer