If a hyperbola has asymptotes \(3x-4y-1=0\) and \(4x-3y-6=0\), then the transverse and conjugate axes of that hyperbola are
Show Hint
- \(x+y-5=0, x-y-1=0\)
- \(4x-3y=0, 3x+4y=0\)
- \(3x-4y=0, 4x+3y=0\)
- \(x+2y-1=0, 2x-y+1=0\)
The Correct Option is A
Solution and Explanation
Step 1: Understand the properties of asymptotes and axes of a hyperbola.
For a hyperbola, the transverse axis and conjugate axis are the bisectors of the angle between the asymptotes.
Let the equations of the asymptotes be \(L_1 = 3x - 4y - 1 = 0\) and \(L_2 = 4x - 3y - 6 = 0\).
The equations of the angle bisectors of two lines \(L_1 = a_1x + b_1y + c_1 = 0\) and \(L_2 = a_2x + b_2y + c_2 = 0\) are given by:
\[ \frac{a_1x + b_1y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2x + b_2y + c_2}{\sqrt{a_2^2 + b_2^2}} \] Step 2: Calculate the denominators for the bisector equations.
For \(L_1 = 3x - 4y - 1 = 0\):
\(\sqrt{a_1^2 + b_1^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
For \(L_2 = 4x - 3y - 6 = 0\):
\(\sqrt{a_2^2 + b_2^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\).
Step 3: Find the equations of the angle bisectors.
Substitute the values into the bisector formula: \[ \frac{3x - 4y - 1}{5} = \pm \frac{4x - 3y - 6}{5} \] Multiply by 5: \[ 3x - 4y - 1 = \pm (4x - 3y - 6) \] Case 1: Using the '+' sign}
\(3x - 4y - 1 = 4x - 3y - 6\)
Rearrange terms:
\(0 = 4x - 3x - 3y + 4y - 6 + 1\)
\(0 = x + y - 5\)
So, one axis is \(x + y - 5 = 0\).
Case 2: Using the '-' sign}
\(3x - 4y - 1 = -(4x - 3y - 6)\)
\(3x - 4y - 1 = -4x + 3y + 6\)
Rearrange terms:
\(3x + 4x - 4y - 3y - 1 - 6 = 0\)
\(7x - 7y - 7 = 0\)
Divide by 7:
\(x - y - 1 = 0\).
So, the equations of the angle bisectors are \(x + y - 5 = 0\) and \(x - y - 1 = 0\). These two lines represent the transverse and conjugate axes of the hyperbola.
Step 4: Verify with the options.
The derived equations are \(x+y-5=0\) and \(x-y-1=0\). This exactly matches Option (1).
Top Questions on Geometry
- The radius and the height of a right circular solid cone are measured as 7 feet each. If there is an error of 0.002 ft for every feet in measuring them, then the error in the total surface area of the cone (in sq. ft) is
- If the volume of a sphere is increasing at the rate of 12 \( \text{cm}^3/\text{sec} \), then the rate (in \( \text{cm}^2/\text{sec} \)) at which its surface area is increasing when the diameter of the sphere is 12 cm is:
- If the lengths of the tangent, subtangent, normal, and subnormal for the curve \( y = x^2 + x - 1 \) at the point \( (1,1) \) are \( a, b, c, \) and \( d \) respectively, then their increasing order is:
- Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) be such that \( x = 0 \) is the only real root of \( P'(x) = 0 \). If \( P(-1)<P(1) \), then in the interval \( [-1,1] \):
- P and Q are the ends of a diameter of the circle \( x^2+y^2=a^2(a>\frac{1}{\sqrt{2}}) \). \( s \) and \( t \) are the lengths of the perpendiculars drawn from P and Q onto the line \( x+y=1 \) respectively. When the product \( st \) is maximum, the greater value among \( s, t \) is:
Questions Asked in AP EAPCET exam
- The sequence of reagents required to convert aniline to benzoic acid is:
- AP EAPCET - 2025
- Reaction mechanisms
- In a binomial distribution, if \(n=4\) and \( P(X=0) = \frac{16}{81} \), then \( P(X=4) = \)
- AP EAPCET - 2025
- Poisson distribution
- If $\sinh^{-1}(x) = \log 3$ and $\cosh^{-1}(y) = \log\left(\frac{3}{2}\right)$, then $\tanh^{-1}(x - y) =$
- AP EAPCET - 2025
- Hyperbolic Functions
- $L_1 = ax - 3y + 5 = 0$ and $L_2 = 4x - 6y + 8 = 0$ are two parallel lines. If $p, q$ are the intercepts made by $L_1 = 0$ and $m, n$ are the intercepts made by $L_2 = 0$ on the X and Y coordinate axes, respectively, then the equation of the line passing through the points $(p, q)$ and $(m, n)$ is
- AP EAPCET - 2025
- Coordinate Geometry
- If $x^2 + y^2 + \sin y = 4$, then the value of $\frac{d^2y}{dx^2}$ at $x=-2$ is
- AP EAPCET - 2025
- Derivatives
