Asymptotes

Q.

If the pair of lines $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ intersect on the $y$-axis, then

1. $2fgh=b{g}^2+c{h}^2$

2. $b{g}^2\ne c{h}^2$

3. $abc=2fgh$

4. None of these

Q.

The curve described para metrically by $x=t^2+2t-1$, $y=3t+5$ represents

1. an ellipse

2. a hyperbola

3. a parabola

4. a circle

Q. If one end of the diameter of a circle is (3, 4) which touches the x-axis then the locus of other end of the diameter of the circle is

1. Circle
2. parabola
3. ellipse
4. hyperbola

Q.

The equations $y=\pm \sqrt{3}x,y=1$ are the sides of

1. an equilateral triangle

2. a right-angled triangle

3. an isosceles triangle

4. a scalene triangle

Q.

if the point $(2,3)$ lies on $k{x}^2–3y^2+2x+y–2=0$ then $k$ is equal to

1. $\frac{1}{17}$

2. $16$

3. $7$

4. $12$

Q. If a hyperbola passes through (2, 3) and has asymptotes 3x - 4y + 5 = 0 and 12x + 5y - 40 = 0, then the equation of its transverse axis is

1. 77x - 21y - 265 = 0
2. 21x - 77y + 265 = 0
3. 21x - 77y - 265 = 0
4. 21x + 77y - 265 = 0

Q. let A ={4, 5, 7} and B={2, 4, 5} be two sets and let a relation R be a relation from A to B is defined by R :{(x, y)/ x>y x belongs to A y belongs to B} then hte difference between the sum of elemetns of domain and range of R is

Q.

$x=1+a+a^2+.....\infty ,\left|a\right|<1$ , $y=1+b+b^2+.......\infty ,\left|b\right|<1$ then the value of $1+ab+a^2{b}^2+.......\infty$

1. $\frac{xy}{\left(x+y-1\right)}$

2. $\frac{xy}{\left(x+y+1\right)}$

3. $\frac{xy}{\left(x-y-1\right)}$

4. $\frac{xy}{\left(x-y+1\right)}$

Q. If a hyperbola passing through the origin has $3x-4y-1=0$ and $4x-3y-6=0$ as its asymptotes, then the equation of its transverse and conjugate axes respectively are

1. $x+y=5$ and $x-y=5$
2. $x+y=5$ and $x-y=1$
3. $x-y=1$ and $x+y=5$
4. $x-y=5$ and $x+y=5$

Q. Find the area of the triangle formed by the lines
(i) y = m₁ x + c₁, y = m₂ x + c₂ and x = 0
(ii) y = 0, x = 2 and x + 2y = 3.
(iii) x + y − 6 = 0, x − 3y − 2 = 0 and 5x − 3y + 2 = 0

Q. Let R be a relation on N defined by x + 2y = 8. The domain of R is

(a) [2, 4, 8]
(b) [2, 4, 6, 8]
(c) [2, 4, 6]
(d) [1, 2, 3, 4]

Q. In a hyperbola portion of tangent intercepted between asymptotes is bisected at the point of contact. Consider a hyperbola whose centre is at origin. A line $x+y=2$ touches this hyperbola at $P(1,1)$ and intersects the asymtotes at $A$ and $B$ such that $AB=6\sqrt{2}$ units.

Angle subtended by $AB$ at centre of the hyperbola is

1. ${\sin }^{-1}\frac{4}{5}$
2. ${\sin }^{-1}\frac{2}{5}$
3. ${\sin }^{-1}\frac{3}{5}$
4. none of these

Q.

Which of the following statements are correct?

1. The particular kind of hyperbola in which the lengths of the transverse and conjugate axis are equal is called an equilateral hyperbola.

2. Eccentricity of equilateral hyperbola = $\sqrt{2}$
3. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=0$

4. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=-a^2$

1. only 1 & 2

2. only 1, 2 & 3

3. only 2, 3 & 4

4. All of them

Q. The eccentricity the hyperbola $x=\frac{a}{2}\left(t+\frac{1}{t}\right),y=\frac{a}{2}\left(t-\frac{1}{t}\right)$ is
(a) $\sqrt{2}$
(b) $\sqrt{3}$
(c) $2\sqrt{3}$
(d) $3\sqrt{2}$

Q. Find the equations to the straight lines which pass through the origin and are inclined at an angle of 75° to the straight line $x+y+\sqrt{3}\left(y-x\right)=a$.

Q. If foci of hyperbola lie on y = x and one of the asymptotes is y = 2x, then equation of the hyperbola, given that it passes through (3, 4), is

1. $x^2-y^2-\frac{5}{2}xy+5=0$
2. $2x^2-2y^2+5xy+5=0$
3. $2x^2+2y^2-5xy+10=0$
4. None of these

Q. Let R be a relation in N defined by (x, y) ∈ R ⇔ x + 2y =8. Express R and R⁻¹ as sets of ordered pairs.

Q. If two tangents can be drawn to the different branches of hyperbola
$\frac{x^2}{1}-\frac{y^2}{4}=1$ from the paint $(\alpha ,{\alpha }^2)$, then

1. $\alpha ϵ(-2,0)$
   
2. $\alpha ϵ(-3,0)$
3. $\alpha ϵ(-\infty ,-2)$
   
4. $\alpha ϵ(-\infty ,-3)$

Q. Find the angles between each of the following pairs of straight lines:
(i) 3x + y + 12 = 0 and x + 2y − 1 = 0
(ii) 3x − y + 5 = 0 and x − 3y + 1 = 0
(iii) 3x + 4y − 7 = 0 and 4x − 3y + 5 = 0
(iv) x − 4y = 3 and 6x − y = 11
(v) (m² − mn) y = (mn + n²) x + n³ and (mn + m²) y = (mn − n²) x + m³.

Q. To what point should the origin be shifted so that the equation x² + xy − 3x − y + 2 = 0 does not contain any first degree term and constant term?

Q.

The equations of the asymptotes of the hyperbola $2x^2+5xy+2y^2-11x-7y-4=0$ are

1. $2x^2+5xy+2y^2-11x-7y-5=0$

2. $2x^2+4xy+2y^2-7x11y+5=0$

3. $2x^2+5xy+2y^2-11x-7y+5=0$

4. none of the above

Q. The product of the perpendicular from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes, is equal to

1. $\frac{ab}{a+b}$
2. $\frac{a^2{b}^2}{a^2+b^2}$
3. $\frac{2a^2{b}^2}{a^2+b^2}$
4. None of these

Q. If $P_1$ and $P_2$ are the perpendiculars from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ on its asymptotes, then :

1. $P_1{P}_2=a^2+b^2$
2. $P_1+P_2=ab$
3. $P_1+P_2=(ab{)}^2$
4. $\frac{1}{P_1{P}_2}=\frac{1}{a^2}+\frac{1}{b^2}$

Q. The center of a rectangular hyperbola lies on the line $y=2x$. If one of the asymptotes is $x+y+c=0$, then the other asymptote is

1. $x-y-3c=0$
2. $2x-y+c=0$
3. $x-y-c=0$
4. none of these

Q. The equations of the assymptotes of the hyperbola $3x^2+10\mathrm{xy}+8y^2+14x+22y+7=0$ are .

1. $3x^2+10\mathrm{xy}+8y^2+14x+22y-7=0$
2. $3x^2+10\mathrm{xy}+8y^2+14x+22y=0$
3. $3x^2+10\mathrm{xy}+8y^2+14x+22y+14=0$
4. $3x^2+10\mathrm{xy}+8y^2+14x+22y+15=0$

Q. if sum of the slopes of the lines given by 4x2-2IkIxy-7y2=0 is equal to the product of the slopes then k=

Q.

Graph each equation. Decide whether each graph represents a linear or nonlinear function

$$\begin{gathered} \mathrm{h}=5+6\mathrm{t} \\ \mathrm{h}=5+6{\mathrm{t}}^2 \end{gathered}$$