Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Conic Section

Multiple Choice Questions

181.

If $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b)$ and x² - y² = c² cut at right angles, then

a² + b² = 2c²
b² - a² = 2c²
a² - b² = 2c²
a²b² = 2c²

182.

The equation of the conic with focus at (1, - 1) directrix along x - y + 1 = 0 and with eccentricity $\sqrt{2}$ , is

x² - y² = 1
xy = 1
2xy - 4x + 4y + 1 = 0
2xy + 4x - 4y - 1 = 0

183.

The number of common tangents to the circles x² + y² = 4 and x² + y² - 6x - 8y = 24 is

184.

The locus of the mid-points of the focal chord of the parabola y² = 4ax is

y² = a(x - a)
y² = 2a(x - a)
y² = 4a(x - a)
None of these

185.

A rod of length l slides with its ends on two perpendicular lines. Then, the locus of its mid point is

$x^{2} + y^{2} = \frac{l^{2}}{4}$
$x^{2} + y^{2} = \frac{l^{2}}{2}$
$x^{2} - y^{2} = \frac{l^{2}}{4}$
None of these

186.

The line joining (5, 0) to ( $10 \cos (θ), 10 \sin (θ)$ ) is divided internally in the ratio 2 : 3 at P. If 0 varies, then the locus of P is

a straight line
a pair of straight lines
a circle
None of the above

187.

If 2x + y + k = 0 is a normal to the parabola y² = - 8x, then the value of k, is

188.

If the equation of an ellipse is 3x² + 2y² + 6x - 8y + 5 = 0, then which of the following are true?

e = $\frac{1}{\sqrt{3}}$
centre is (- 1, 2)
foci are (- 1, 1) are (- 1, 3)
All of the above

189.
The equation of the common tangents to the two hyperbolas $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 and \frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1, are$

$y = \pm x \pm \sqrt{b^{2} - a^{2}}$

$y = \pm x \pm \sqrt{a^{2} - b^{2}}$

$y = \pm x \pm \sqrt{a^{2} + b^{2}}$

$y = \pm x \pm (a^{2} - b^{2})$

$y = \pm x \pm \sqrt{a^{2} - b^{2}}$

We have the hyperbolas

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ...(i)

and $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$ ...(ii)

Any tangent to the hyperbola Eq. (i),

y = mx + c

where c = $\pm \sqrt{a^{2} m^{2} - b^{2}}$ ...(iii)

But this tangent touches the parabola Eq. (ii), also

$∴ \frac{{(mx + c)}^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 \Rightarrow b^{2} (m^{2} x^{2} + c^{2} + 2 cmx) - a^{2} x^{2} = a^{2} b^{2} \Rightarrow (b^{2} m^{2} - a^{2}) x^{2} + 2 {mcxb}^{2} + b^{2} c^{2} - a^{2} b^{2} = 0 \Rightarrow (b^{2} m^{2} - a^{2}) x^{2} + 2 {mcb}^{2} x + b^{2} (c^{2} - a^{2}) = 0 For the tangency, it should have equal roots {(2 {mcb}^{2})}^{2} = 4 (b^{2} m^{2} - a^{2}) . b^{2} (c^{2} - a^{2})$

$\Rightarrow 4 m^{2} c^{2} b^{4} = 4 b^{2} (b^{2} m^{2} c^{2} - b^{2} m^{2} a^{2} - a^{2} c^{2} + a^{4}) \Rightarrow m^{2} c^{2} b^{2} = b^{2} m^{2} c^{2} - b^{2} m^{2} a^{2} - a^{2} c^{2} + a^{4} \Rightarrow a^{2} c^{2} = a^{4} - b^{2} m^{2} a^{2} \Rightarrow c^{2} = a^{2} - b^{2} m^{2} \Rightarrow a^{2} m^{2} - b^{2} = a^{2} - b^{2} m^{2} [∵ using Eq . (iii)] \Rightarrow (a^{2} + b^{2}) m^{2} = a^{2} + b^{2} \Rightarrow m^{2} = 1 \Rightarrow m = \pm 1$