Name: Condition for a Line to be a Tangent of Hyperbola
Uploaded: 2019-07-19
Duration: 6 min 28 s
Description: How to determine whether a given line is tangent to a hyperbola or not? Watch this amazing video to explore everything about a tangent to a hyperbola.

Question

Find the equations of tangent drawn to hyperbola $2 x^{2} - 3 y^{2} = 6 through (- 2, 1)$ .

Accepted Answer

Equation of hyperbola is $2 x^{2} - 3 y^{2} = 6$ .

$\Rightarrow \frac{x^{2}}{3} - \frac{y^{2}}{2} = 1$

Let $m$ be the slope of the tangent.

The tangent is passing through $P (- 2, 1)$ .

Equation of tangent is $y - 1 = m (x + 2)$

$\Rightarrow y - 1 = m x + 2 m$

$\Rightarrow y = m x + (2 m + 1) \dots (1)$

Since $(1)$ is a tangent to hyperbola so using condition of tangency we have:

$c^{2} = a^{2} m^{2} - b^{2}$

$\Rightarrow (2 m + 1)^{2} = 3 m^{2} - 2 (∵ a^{2} = 3, b^{2} = 2, c = 2 m + 1)$

$\Rightarrow 4 m^{2} + 4 m + 1 = 3 m^{2} - 2$

$\Rightarrow m^{2} + 4 m + 3 = 0$

$\Rightarrow m^{2} + 3 m + m + 3 = 0$

$\Rightarrow m (m + 3) + 1 (m + 3) = 0$

$\Rightarrow (m + 1) (m + 3) = 0$

$\Rightarrow m = - 1 or - 3$ .

Case $1)$ When $m = - 1$

Equation of tangent is:

$y = - x - 1$

$\Rightarrow x + y + 1 = 0$

Case $2)$ When $m = - 3$

Equation of tangent is:

$y = - 3 x - 5$

$\Rightarrow 3 x + y + 5 = 0$ .

Find the equations of tangent drawn to hyperbola $2 x^{2} - 3 y^{2} = 6 through (- 2, 1)$ .

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

Find the equations of tangent drawn to hyperbola 2x2−3y2=6 through (−2,1).

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

Find the equations of tangent drawn to hyperbola $2 x^{2} - 3 y^{2} = 6 through (- 2, 1)$ .