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Prove that the straight line lx+my+n=0 t...

Prove that the straight line `lx+my+n=0` touches the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` if `a^(2)l^(2)-b^(2)m^(2)=n^(2)`.

A

`a^(2)l^(2)+b^(2)m^(2)=n^(2)`

B

`a^(2)l^(2)-b^(2)m^(2)=n^(2)`

C

`a^(2)m^(2)-b^(2)n^(2)=l^(2)`

D

`a^(2)n^(2)-b^(2)l^(2)=m^(2)`

Text Solution

Verified by Experts

The equation of the line is
`lx+my+n=0impliesy=(-(l)/(m))x+(-(n)/(m))`……`(i)`
We know that the line `y=mx+c` touches the hyperbola
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` if `c^(2)=a^(2)m^(2)-b^(2)`.
So, line `(i)` will touch `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`, if
`(-(n)/(m))^(2)=a^(2)(-(l)/(m))^(2)-b^(2)impliesn^(2)=a^(2)l^(2)-b^(2)m^(2)`
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OBJECTIVE RD SHARMA ENGLISH-HYPERBOLA-Chapter Test
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  2. Find the value of m for which y=m x+6 is tangent to the hyperbola (x^2...

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  3. The equation of the tangent to the hyperbola 4y^(2)=x^(2)-1 at the poi...

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  4. The number of normals to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))...

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  5. If e and e1 are the eccentricities of the hyperbola xy=c^(2) and x^(2)...

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  6. A rectangular hyperbola with centre C, is intersect by a circle of rad...

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  7. The equation of the pair of asymptotes of the hyperbola xy-4x+3y=0, is

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  8. If the latus rectum of the hyperbola (x^(2))/(16)-(y^(2))/(b^(2))=1 is...

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  9. Chords of the hyperbola x^(2)-y^(2)=a^(2) touch the parabola y^(2)=4ax...

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  10. Tangents drawn from the point (c, d) to the hyperbola (x^(2))/(a^(2))-...

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  11. If the tangent at (h, k) on b^2x^2-a^2y^2=a^2b^2 cuts the auxiliary ci...

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  12. If the chords of contact of tangents drawn from P to the hyperbola x^(...

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  13. The tangent at a point P on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(...

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  14. The mid-point of the chord intercepted by the hyperbola 9x^(2)-16y^(2)...

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  15. Locus of P such that the chord of contact of P with respect to y^2=4ax...

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  16. C is the center of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 The tangen...

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  17. If lx+my+n=0 is a tangent to the rectangular hyperbola xy=c^(2), then

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  18. A tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 cuts the ellipse ...

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  19. The product of lengths of perpendicular from any point on the hyperbol...

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  20. The angle between the asymptotes of the hyperbola 3x^(2)-y^(2)=3, is

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  21. Find the area of the triangle formed by any tangent to the hyperbola (...

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