Angle of Intersection of Two Curves

Q.

Is slope of a line is defined as the tangent of the angle it makes with the positive x axis or simply the x- axis I.e. both positive and negative

Q. If the curves, $\frac{x^2}{a}+\frac{y^2}{b}=1$ and $\frac{x^2}{c}+\frac{y^2}{d}=1$ intersect each other at an angle of ${90}^{\circ }$, then which of the following relations is $\text{TRUE}?$

1. $a+b=c+d$
2. $a-b=c-d$
3. $ab=\frac{c+d}{a+b}$
4. $a-c=b+d$

Q. $P_1,P_2$ are points on either of the two lines $y-\sqrt{3}|x|=2$ at a distance of $5$ units from their point of intersection. Find the coordinates of the foot of perpendiculars drawn from $P_1,P_2$ on the bisector of the angle between the given lines.

Q. If the curves $x=y^4$ and $xy=k$ cut at right angles, then $(4k{)}^6$ is equal to

Q.

If the curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{{\alpha }^2}+\frac{y^2}{{\beta }^2}=1$ cut each other orthogonally,

1. $a^2+b^2={\alpha }^2+{\beta }^2$

2. $a^2-b^2={\alpha }^2-{\beta }^2$
   

3. $a^2-b^2={\alpha }^2+{\beta }^2$
   

4. $a^2+b^2={\alpha }^2-{\beta }^2$

Q.

The vertices of a triangle are $A(3,7),B(3,4)$ and $C(5,4)$.

The equation of the bisector of $\angle ABC$ is

1. $y=x+1$

2. $y=x-1$

3. $y=3x-5$

4. $y=x$

Q.

The angle of intersection of the curves :

xy = 6 and x²y = 12 is ... .

1. 

2. 

3. 

4. 

Q.

The angle of intersection of the curves $y^2=\frac{2x}{\pi }andy=sinx$ is

1. 

2. 

3. 

4. 

Q. The acute angle between the curves $y^2=x\mathrm{and}{x}^2=y$ at (1, 1) is .

1. ${\tan }^{-1}\frac{4}{3}$
2. ${\tan }^{-1}\frac{3}{4}$
3. $90°$
4. $45°$

Q.

The angle of intersection of the curves $y=2si{n}^{2}xandy=cos2xatx=\frac{\pi }{6}$ is

1. π/4

2. π/3

3. π/2

4. None of these

Q. Which of the following pair of curves is/are orthogonal?
(where $c$ and $k$ arbitrary constant)

1. $16x^2+y^2=c$ and $y^{16}=kx$
2. $y=x+c{e}^{-x}$ and $x+2=y+k{e}^{-y}$
3. $y=c{x}^2$ and $x^2+2y^2=k$
4. $x^2-y^2=c$ and $xy=k$

Q.

Find the slope of the line, which makes an angle of 30° with the positive direction of y-axis measured anticlockwise.

Q. Which of the following pairs is/are orthogonal?
($c$ and $k$ are arbitrary constants.)

1. $16x^2+y^2=c$ and $y^{16}=kx$
2. $y=x+c{e}^{-x}$ and $x+2=y+k{e}^{-y}$
3. $y=c{x}^2$ and $x^2+2y^2=k$
4. $x^2-y^2=c$ and $xy=k$

Q.

Show that the normal at any point θ to the curve

is at a constant distance from the origin.

Q.

The two curves $x^3-3x{y}^2+2=0and3x^{2}y-y^3-2=0$

1. Cut at an angle π/3

2. Touch each other

3. Cut at an angle π/4

4. Cut at right angles

Q. Family of curves which makes an angle $\frac{\pi }{4}$ whith the family of hyperbola $xy=c^2$ is

1. $x^2-4xy-y^2=k$
   
   
2. $x^2+2xy-y^2=k$
   
3. $x^2-2xy-y^2=k$
   
4. $x^2+xy-y^2=k$

Q. Let $\theta$ be the acute angle between the curves $y=|x^2-2|$ and $y=\sqrt{8-x^2}$ at their point(s) of intersection. Then the value of $\tan \theta$ is

1. $0$
2. $\frac{5}{3}$
3. $\frac{3}{5}$
4. $1$

Q. The two curves $x^3-3x{y}^2+2=0$ and $3x^{2}y-y^3=2$

1. touch each other
2. cut at an angle $\frac{\pi }{4}$
3. cut each other at right angle
4. cut at an angle $\frac{\pi }{3}$

Q. sin18^°sin42^°sin78^°csc54^°=?

Q. If $x+4y=14$ is a normal to the curve $y^2=\alpha x^3-\beta$ at the point $(2,3),$ then the value of $\alpha +\beta$ is

Q. The lines tangent to the curve $x^3+2y^3+3x^{2}y-2y{x}^2+3x-2y=0\text{and}{x}^7-y^4+2x+3y=0$ at the origin intersect at an angle $\theta$, then the value of $\theta$ is equal to

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q.

The curves x²+y²=16 and y²=6x intersects at

1. (0, 2)

2. (2, 2√3)

3. (2, 0)

4. (0, 2√3)

Q. Angle between the curves $x^{2}y=1-y$ and $x^3=2(1-y)$ is

1. ${\tan }^{-1}4$
2. ${\tan }^{-1}\frac{3}{4}$
3. ${\tan }^{-1}\frac{4}{7}$
4. $\frac{\pi }{4}$

Q. The two parabolas $y^2=4x$ and $x^2=4y$ intersect at a point P, whose abscissae is not zero, such that

1. they both touch each other at P
2. they cut at right angles at P
3. the tangents to each curve at P make complementary angles with the x-axis
4. none of these

Q.

The angle of intersection of curves. y = [|sin x| + |cos x|] and $x^2+y^2=5$ where [.] denotes greatest integral function is

1. $\frac{\pi }{4}$

2. $ta{n}^{-1}\left(\frac{1}{2}\right)$

3. $ta{n}^1\left(2\right)$

4. none of these

Q. Find the point on the curve $9y^2=x^3$ while normal to the curve makes equal intercepts with the axes

1. $(4,\frac{8}{3})$
2. $(4,\frac{5}{3})$
3. $(2,\frac{8}{3})$
4. $(4,\frac{7}{3})$

Q. State the following statement is True or False

1. True
2. False

Q. Let $\theta$ be the acute angle between the curves $y=|x^2-2|$ and $y=\sqrt{8-x^2}$ at their point(s) of intersection. Then the value of $\tan \theta$ is

1. $0$
2. $\frac{3}{5}$
3. $\frac{5}{3}$
4. $1$

Q. if α, ß, Γ and δ be the eccentric angles of the feet of four co-normal points of a hyperbola x²/a²-y²/b²=1 from any point in its plane then α+ß+ Γ+ δ can be

Q. The equation of line passing through $(2,-3,5)$ and making congruent angles with all three co-ordinate axes is _______ ($k\in R$)

1. $F=(2,-3,5)+k(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}),k\in R$
2. $F=(2,-3,5)+k(1,1,1),k\in R$
3. $F=(2,-3,5)+k(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}),k\in R$
4. $F=(2,-3,5)+k(-1,1,1),k\in R$