If {m_1} and {m_2} are the slopes of the tang | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The equation of a line passing through $(6, 2)$ with slope $m$ is $y - 2 = m(x - 6)$,which simplifies to $y = mx + (2 - 6m)$.For this line to be a

Vedclass · Accepted Answer

Both $(A)$ and $(B)$

Vedclass · Answer

(D) The equation of a line passing through $(6, 2)$ with slope $m$ is $y - 2 = m(x - 6)$,which simplifies to $y = mx + (2 - 6m)$.For this line to be a tangent to the hyperbola $\frac{x^2}{25} - \frac{y^2}{16} = 1$,the condition of tangency $c^2 = a^2m^2 - b^2$ must be satisfied,where $c = 2 - 6m$,$a^2 = 25$,and $b^2 = 16$.Substituting these values: $(2 - 6m)^2 = 25m^2 - 16$.Expanding the left side: $4 + 36m^2 - 24m = 25m^2 - 16$.Rearranging the terms: $11m^2 - 24m + 20 = 0$.Since ${m_1}$ and ${m_2}$ are the roots of this quadratic equation,by Vieta's formulas:Sum of roots: ${m_1} + {m_2} = -(\frac{-24}{11}) = \frac{24}{11}$.Product of roots: ${m_1}{m_2} = \frac{20}{11}$.Thus,both $(A)$ and $(B)$ are correct.

If ${m_1}$ and ${m_2}$ are the slopes of the tangents to the hyperbola $\frac{x^2}{25} - \frac{y^2}{16} = 1$ which pass through the point $(6, 2)$,then:

Similar Questions

If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the coordinate axes at the distinct points $A$ and $B$,then the locus of the midpoint of $AB$ is

If $e_{1}$ and $e_{2}$ represent the eccentricity of the curves $16x^{2}-9y^{2}=144$ and $9x^{2}-16y^{2}=144$ respectively,then $\frac{1}{e_{1}^{2}}+\frac{1}{e_{2}^{2}}$ is equal to

If $l$ is the maximum value of $-3x^2+4x+1$ and $m$ is the minimum value of $3x^2+4x+1$,then the equation of the hyperbola having foci at $(l, 0)$ and $(7m, 0)$ and eccentricity $e=2$ is

If the vertices and foci of a hyperbola are respectively $(\pm 3,0)$ and $(\pm 4,0)$,then the parametric equations of that hyperbola are:

The angle between the asymptotes of the hyperbola $x^2-3y^2=3$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module