Slope Formula for Angle of Intersection of Two Curves

Q. Consider the curve $x=1-3t^2,y=t-3t^3$. The tangent at any point on the curve is inclined at an angle $\theta$ with the positive x-axis. If tangent at point $P(-2,2)$ cuts the curve again at point $Q$, then

1. $\tan \theta \pm \sec \theta =3t$
2. Coordinates of $Q$ is $(-\frac{1}{3},-\frac{2}{3})$
3. Angle between tangents at $P$ and $Q$ is $\frac{\pi }{2}$
4. Angle between tangents at $P$ and $Q$ is $\frac{\pi }{4}$

Q.

The angle between the pair of straight lines formed by joining the points of intersection of $x^2+y^2=4$and $y=3x+c$ to the origin is a right angle. Then, $c^2$ is equal to

1. $20$

2. $13$

3. $\frac{1}{5}$

4. $5$

Q. Tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ make angles ${\theta }_1,{\theta }_2$ with the transverse axis. If ${\theta }_1,{\theta }_2$ are
complementary then the locus of the point of intersection of the tangents is

1. 
   
2. 
   
   
3. 
   
   
4. 
   
   

Q. Let $L{L}^{{}^{\prime }}$ be the latus rectum through the focus of the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $A^{{}^{\prime }}$ be the farther vertex. If
$\Delta A^{{}^{\prime }}L{L}^{{}^{\prime }}$ is equilateral, then the eccentricity of the hyperbola is

1. 
   
2. 
   
3. 
   
4. 

Q. If the tangent to the curve, $y=f\left(x\right)=x{\log }_{e}x,$ $(x>0)$ at a point $(c,f(c\left)\right)$ is parallel to the line-segment joining the points $(1,0)$ and $(e,e)$, then $c$ is eqaul to

1. $e^{\left(\frac{1}{1-e}\right)}$
   
2. $\frac{e-1}{e}$
   
3. $\frac{1}{e-1}$
   
4. $e^{\left(\frac{1}{e-1}\right)}$
   

Q. The angle of intersection of the curves $y=x^2$and $x=y^2$ at (1, 1) is
[Roorkee 2000; Karnataka CET 2001]

1. $ta{n}^{-1}\left(\frac{4}{3}\right)$
2. $ta{n}^{-1}\left(1\right)$
3. ${90}^o$
4. $ta{n}^{-1}\left(\frac{3}{4}\right)$

Q. If from a point $P$ representing the complex number $z_1$ on the curve $|z|=2$, two tangents are drawn from $P$ to the curve $|z|=1$, meeting at points $Q\left(z_2\right)$ and $R\left(z_3\right)$, then

1. $(\frac{4}{\left(\overline{z_1}\right)}+\frac{1}{(\overline{z_2)}}+\frac{1}{\left(\overline{z_3}\right)})(\frac{4}{\left(\overline{z_1}\right)}+\frac{1}{(\overline{z_2)}}+\frac{1}{\left(\overline{z_3}\right)})=9$
2. Complex number $(z_1+z_2+z_3)/3$ will be on the curve $|z|=1$
3. $arg\left(\frac{z_2}{z_3}\right)=\frac{2\pi }{3}$
4. Orthocentre and circumcenter of $\Delta PQR$ will coincide

Q. The angle between curves $y^2=4x$ and $x^2+y^2=5$ at (1, 2) is
[Karnataka CET 1999]

1. $ta{n}^{-1}\left(3\right)$
2. $ta{n}^{-1}\left(2\right)$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{4}$

Q.

Which of the following gives the length of subtangent ?

[PT = length of tangent, (x1, y1) is the point where the tangent is drawn, $\theta$ is the angle the tangent makes with x-axis, x1, y1, PT > 0]

1. 

2. PT cos θ

Q. Two straight lines rotate about two fixed points. If they start from their position of coincidence such that one rotates at the rate double that of the other. Prove that the locus of their point of intersection is a circle.

Q. If the tangent to the curve $y=\frac{x}{x^2-3},x\in R,(x\ne \pm \sqrt{3},)$ at a point $(\alpha ,\beta )\ne (0,0)$ is parallel to the line $2x+6y-11=0,$ then $|6\alpha +2\beta |=$

Q. The function $f\left(x\right)=(x^2-1)|x^2-3x+2|+\cos \left(\left|x\right|\right)$ is NOT differentiable at:

1. $-1$
2. $0$
3. $1$
4. $2$

Q. Let $P(h,k)$ be a point on the curve $y=x^2+7x+2,$ nearest to the line, $y=3x-3$. Then the equation of the normal to the curve at $P$ is:

1. $x+3y-62=0$
2. $x-3y-11=0$
3. $x-3y+22=0$
4. $x+3y+26=0$

Q. The angle between curves $y^2=4x$ and $x^2+y^2=5$ at (1, 2) is
[Karnataka CET 1999]

1. $ta{n}^{-1}\left(3\right)$
2. $ta{n}^{-1}\left(2\right)$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{4}$

Q.

f(x) and g(x) intersect at (3, 5). If given that f’(3) = 7 and g’(3) = -1 then find the angle of intersection between f(x) and g(x) at (3, 5)

1. $\theta =ta{n}^{-1}\left(4/3\right)$
2. $\theta =ta{n}^{-1}\left(3/4\right)$
3. $\theta =ta{n}^{-1}\left(5/3\right)$
4. None of these

Q. $\alpha ,\beta$ are the angles made by the tangents with the axis of the parabola drawn from the point $P$ to the paraboala $y^2=4ax$

List - I	List - II
A. $\cot \alpha \cot \beta =k$ , locus of P is	1. $kx=a$
B. $\tan \alpha +\tan \beta =k$ , locus of P is	2. $y=k(x-a)$
C. $\tan (\alpha +\beta )=k$ locus of P is	3. $kx=y$
D. $\tan \alpha \tan \beta =k$ , locus of P is	4. $xy=k$
	5. $x=ka$

The correct match for List - I from List - II

1. $A-5,B-3,C-1,D-2$
   
2. $A-1,B-3,C-2,D-4$
3. $A-3,B-2,C-1,D-5$
4. $A-4,B-2,C-1,D-5$

Q. Find the angle between the lines joining the points (-1, 2), (3, -5) and (-2, 3), (5, 0).

Q. The angle between the tangent at any point $P$ and the line joining $P$ to the origin, where $P$ is a point on the curve $\ln (x^2+y^2)=c{\tan }^{-1}\frac{y}{x},c$ is constant, is

1. Independent of $x$
2. Independent of $y$
3. Independent of $x$ but dependent on $y$
4. Independent of $y$ but dependent on $x$

Q.

f(x) and g(x) intersect at (3, 5). If given that f’(3) = 7 and g’(3) = -1 then find the angle of intersection between f(x) and g(x) at (3, 5)

1. $\theta =ta{n}^{-1}\left(3/4\right)$
2. $\theta =ta{n}^{-1}\left(4/3\right)$
3. $\theta =ta{n}^{-1}\left(5/3\right)$
4. None of these

Q. The angle of intersection of the curves $y=x^2$and $x=y^2$ at (1, 1) is
[Roorkee 2000; Karnataka CET 2001]

1. $ta{n}^{-1}\left(\frac{4}{3}\right)$
2. $ta{n}^{-1}\left(1\right)$
3. ${90}^o$
4. $ta{n}^{-1}\left(\frac{3}{4}\right)$

Q. The angle between curves $y^2=4x$ and $x^2+y^2=5$ at (1, 2) is
[Karnataka CET 1999]

1. $ta{n}^{-1}\left(3\right)$
2. $ta{n}^{-1}\left(2\right)$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{4}$

Q.

f(x) and g(x) intersect at (3, 5). If given that f’(3) = 7 and g’(3) = -1 then find the angle of intersection between f(x) and g(x) at (3, 5)

1. $\theta =ta{n}^{-1}\left(4/3\right)$
2. $\theta =ta{n}^{-1}\left(3/4\right)$
3. $\theta =ta{n}^{-1}\left(5/3\right)$
4. None of these

Q. The angle between the tangent at any point $P$ and the line joining $P$ to the origin, where $P$ is a point on the curve $\ln (x^2+y^2)=c{\tan }^{-1}\frac{y}{x},c$ is constant, is

1. Independent of $x$
2. Independent of $y$
3. Independent of $x$ but dependent on $y$
4. Independent of $y$ but dependent on $x$

Q. If a vector make angles $\alpha ,\beta$ and $\gamma$ with axes $x,y$ and $z$ , then $\cos 2\alpha +\cos 2\beta +\cos 2\gamma =$

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

Q. The angle of intersection of the curves $y=x^2$and $x=y^2$ at (1, 1) is
[Roorkee 2000; Karnataka CET 2001]

1. $ta{n}^{-1}\left(\frac{4}{3}\right)$
2. $ta{n}^{-1}\left(1\right)$
3. ${90}^o$
4. $ta{n}^{-1}\left(\frac{3}{4}\right)$

Q.

f(x) and g(x) intersect at (3, 5). If given that f’(3) = 7 and g’(3) = -1 then find the angle of intersection between f(x) and g(x) at (3, 5)

1. None of these

2. θ = tan^-1 (3 / 4)

3. θ = tan^-1 (5 / 3)

4. θ = tan^-1 (4 / 3)

Q. Find the lengths of the medians $RS$ and $PT$ of a triangle $PQR$ whose vertices are $P(6,-2),Q(6,3)$ and $R(3,1)$