A hyperbola passing through a focus of the ellipse...

A hyperbola passing through a focus of the ellipse

\frac{x^2}{169}+\frac{y^2}{25}=1

. Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse. The product of eccentricities is 1 . Then, the equation of the hyperbola is

Options

A $\frac{x^2}{144}-\frac{y^2}{9}=1$
B $\frac{x^2}{169}-\frac{y^2}{25}=1$
C $\frac{x^2}{144}-\frac{y^2}{25}=1$
D $\frac{x^2}{25}-\frac{y^2}{9}=1$

Correct answer

C. $\frac{x^2}{144}-\frac{y^2}{25}=1$

Step-by-step solution

Let the equation of hyperbola be

\frac{x^2}{a^2}-\frac{y^2}{b^2}=

1 Given equation of ellipse is

\frac{x^2}{(13)^2}+\frac{y^2}{(5)^2}=

1 Here,

\begin{aligned}& \text { Here, } \quada=13, b=5 \\ & \therefore \quad e=\sqrt{1-\frac{b^2}{a^2}}\\ & \therefore \quad=\sqrt{1-\frac{25}{169}}=\sqrt{\frac{144}{169}}=\frac{12}{13}\\ & \therefore \text { Focus }( \pma e, 0)=\left( \pm13 \times \frac{12}{13},0\right) \\ & \quad=( \pm12,0) \end{aligned}

Since, Eq. (i) passes through

( \pm 12,0)

\begin{array}{rrrl} \therefore& \frac{144}{a^2}-\frac{0}{b^2}=1 \\ \Rightarrow & a²=144 \\ \Rightarrow & a= \pm12 \end{array}

Now eccentricity of hyperbola

\begin{aligned}e^{\prime} & =\sqrt{1+\frac{b^2}{a^2}}\\ & =\sqrt{1+\frac{b^2}{144}} \end{aligned}

According to the equation,

\begin{array}{rlrl}& e^{\prime}=1 \\ \Rightarrow & & \frac{12}{13} \times \sqrt{1+\frac{b^2}{144}}& =1 \\ \Rightarrow & & \sqrt{1+\frac{b^2}{144}}& =\frac{13}{12}\\ \Rightarrow & & 1+\frac{b^2}{144}& =\frac{169}{144}\\ \Rightarrow & & \frac{b^2}{144}& =\frac{169}{144}-1 \\ \Rightarrow & & \frac{b^2}{144}& =\frac{25}{144}\\ b² & =25 \end{array}

\therefore

Equation of hyperbola is

\frac{x^2}{144}-\frac{y^2}{25}=

1

Practice Hyperbola on Quantrex Academy →

More from Hyperbola

If the eccentricity e of the hyperbola (x^2)/(a^2) - (y^2)/(b^2) = 1 , passing through… JEE Main 2026 Let the eccentricity e of a hyperbola satisfy the equation 6e^2 - 11e + 3 = 0 . If the… JEE Main 2026 Let H: (x^2)/(a^2)-(y^2)/(b^2)=1 be a hyperbola such that the distance between its foci… JEE Main 2026 Let O be the origin, and P and Q be two points on the rectangular hyperbola xy = 12 such… JEE Main 2026 For some θ ∈(0, (π)/(2)) , let the eccentricity and the length of the latus rectum of the… JEE Main 2026 Let PQ be a chord of the hyperbola x^ 2 4 - y^ 2 b^ 2 =1 , perpendicular to the x -axis… JEE Main 2026 Let the domain of the function f(x)= _ 3 _ 5 _ 7 (9 x-x^ 2 -13) be the interval (m, n) .… JEE Main 2026 Let P(10,2 15 ) be a point on the hyperbola x^ 2 a^ 2 - y^ 2 ~b^ 2 =1 , whose foci are S… JEE Main 2026 If the line α x+2 y=1 , where α ∈ ℝ , does not meet the hyperbola x^ 2 -9 y^ 2 =9 , then… JEE Main 2026 Let the foci of a hyperbola coincide with the foci of the ellipse x^ 2 36 + y^ 2 16 =1 .… JEE Main 2026 Let e_1 and e_2 be the eccentricities of the ellipse x^2 ~b^2 + y^2 25 =1 and the… JEE Main 2025 Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be… JEE Main 2025 Consider the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 having one of its focus at P(-3,0) . If… JEE Main 2025 Let the sum of the focal distances of the point P(4,3) on the hyperbola H: x^2 a^2 - y^2… JEE Main 2025 If the equation of the hyperbola with foci (4,2) and (8,2) is 3 x^2-y^2-α x+β y+γ=0… JEE Main 2025 Let the product of the focal distances of the point P(4,2 3 ) on the hyperbola H: x^2 a^2… JEE Main 2025 Full Hyperbola list Hyperbola JEE Maths PYQs JEE Mathematics PYQs All 99 Percentile Qs Bank for JEE Main PYQs