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Conic Section

Multiple Choice Questions

251.

A line passing through the point of intersection of x + y = 4 and x - y = 2 makes an angle $\tan^{- 1} (\frac{3}{4})$ with the x-axis. It intersects the parabola y² = 4(x-3) at points $(x_{1}, y_{1}) and (x_{2}, y_{2})$ respectively. Then, $|x_{1} - x_{2}|$

$\frac{16}{9}$
$\frac{32}{9}$
$\frac{40}{9}$
$\frac{80}{9}$

252.

The equation of auxiliary circle of the ellipse 16x² + 25y² + 32x - 100y = 284 is

x² + y² + 2x - 4y - 20 = 0
x² + y² + 2x - 4y = 0
(x + 1)² + (y - 2)² = 400
(x + 1)² + (y - 2)² = 225

253.
If PQ is a double ordinate of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 such that ∆ OPQ$ is equilateral. O being the centre. Then, the eccentricity e satisfies

$1 < e < \frac{2}{\sqrt{3}}$

e = $\frac{2}{\sqrt{2}}$

e = $\frac{\sqrt{3}}{2}$

$e > \frac{2}{\sqrt{3}}$

$e > \frac{2}{\sqrt{3}}$

Given equation of hyperbola is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and $∆$ OPQ is an equilateral triangle. PQ is double ordinate of the hyperbola.

Let the coordinates of P and Q be ( $asec (θ), btan (θ)$ ) and ( $asec (θ), - btan (θ)$ ), respectively.

$In ∆ OPQ, OP = PQ \Rightarrow a^{2} \sec^{2} (θ) + b^{2} \tan^{2} (θ) {(2 btan (θ))}^{2} \Rightarrow a^{2} \sec^{2} (θ) = 3 b^{2} \tan^{2} (θ) \Rightarrow \sin^{2} (θ) = \frac{a^{2}}{3 b^{2}}$

Now, $\sin^{2} (θ) < 1$

$\Rightarrow \frac{a^{2}}{3 b^{2}} < 1 \Rightarrow \frac{b^{2}}{a^{2}} > \frac{1}{3} \Rightarrow 1 + \frac{b^{2}}{a^{2}} > \frac{4}{3} ∴ e^{2} > \frac{4}{3} \Rightarrow e > \frac{2}{\sqrt{3}}$

254.

If the vertex of the conic y - 4y = 4x - 4a always lies between the straight lines x + y = 3 and 2x + 2y - 1 = 0, then

2 < a < 4
$- \frac{1}{2} < a < 2$
0 < a < 2
$- \frac{1}{2} < a < \frac{3}{2}$

255.

The locus of the mid-points of chords of the circle x² + y² = 1, which subtends a right angle at the origin, is

x² + y² = $\frac{1}{4}$
$x^{2} + y^{2} = \frac{1}{2}$
xy = 0
x² - y² = 0

256.

x = - a
x = a
x = 0
x = - $\frac{a}{2}$

257.

The points of the ellipse 16x² + 9y² = 400 at which the ordinate decreases at the same rate at which the abscissa increases is/are given by

$(3, \frac{16}{3}) and (- 3, - \frac{16}{3})$
$(3, - \frac{16}{3}) and (- 3, \frac{16}{3})$
$(\frac{1}{16}, \frac{1}{9}) and (- \frac{1}{16}, - \frac{1}{9})$
$(\frac{1}{16}, - \frac{1}{9}) and (- \frac{1}{16}, \frac{1}{9})$

258.

If the parabola x² = ay makes an intercept of length $\sqrt{40}$ units on the line y - 2x = 1, then a is equal to

259.

If the vertex of the conic y - 4y = 4x - 4a always lies between the straight lines x + y = 3 and 2x + 2y - 1 = 0, then

2 < a < 4
$- \frac{1}{2} < a < 2$
0 < a < 2
$- \frac{1}{2} < a < \frac{3}{2}$

260.

Number of intersecting points of the conics 4x² + 9y² = 1 and 4x² + y² = 4 is

Previous Year Papers

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Multiple Choice Questions

253. If PQ is a double ordinate of the hyperbola x2a2 - y2b2 = 1 such that ∆OPQ is equilateral. O being the centre. Then, the eccentricity e satisfies 1 < e < 23 e = 22 e = 32 e > 23

253.
If PQ is a double ordinate of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 such that ∆ OPQ$ is equilateral. O being the centre. Then, the eccentricity e satisfies

$1 < e < \frac{2}{\sqrt{3}}$

e = $\frac{2}{\sqrt{2}}$

e = $\frac{\sqrt{3}}{2}$

$e > \frac{2}{\sqrt{3}}$