Tangents are drawn to the hyperbola x 2 9 - y 2 4 = 1 , parallel to the straight line 2 x - y = 1 . The points of contact of the tangents on the hyperbola are [2012]

Question

Tangents are drawn to the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1,$ parallel to the straight line $2 x - y = 1$ . The points of contact of the tangents on the hyperbola are [2012]

Smart Achievers · Accepted Answer

(1, 2)

If slope of tangents to hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is $m$ , then equation of tangent to the hyperbola is

$y = m x \pm \sqrt{a^{2} m^{2} - b^{2}}$ with the points of contact $(\frac{\pm a^{2} m}{\sqrt{a^{2} m^{2} - b^{2}}}, \frac{\pm b^{2}}{\sqrt{a^{2} m^{2} - b^{2}}})$

$∵$ $Tangent to hyperbola \frac{x^{2}}{9} - \frac{y^{2}}{4} = 1 is parallel to 2 x - y = 1$

$∴$ $Slope of tangent = 2$

$∴$ $Points of contact are$ $(\frac{\pm 9 \times 2}{\sqrt{9 \times 4 - 4}}, \frac{\pm 4}{\sqrt{9 \times 4 - 4}})$

i.e. $(\frac{9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}})$ and $(\frac{- 9}{2 \sqrt{2}}, \frac{- 1}{\sqrt{2}})$

Solution

Q.
Tangents are drawn to the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1,$ parallel to the straight line $2 x - y = 1$ . The points of contact of the tangents on the hyperbola are [2012]

1 $(\frac{9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}})$

2 $(- \frac{9}{2 \sqrt{2}}, - \frac{1}{\sqrt{2}})$

3 $(3 \sqrt{3}, - 2 \sqrt{2})$

4 $(- 3 \sqrt{3}, 2 \sqrt{2})$

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Solution

Q. Tangents are drawn to the hyperbola x29-y24=1, parallel to the straight line 2x-y=1. The points of contact of the tangents on the hyperbola are [2012]

1 (922,12)

2 (-922,-12)

3 (33,-22)