Conic Section

Q. If a curve passes through the origin and the slope of the tangent to it at any point $(x,y)$ is $\frac{x^2-4x+y+8}{x-2}$, then this curve also passes through the point :

1. $(5,4)$
2. $(4,4)$
3. $(5,5)$
4. $(4,5)$

Q. If $2p^2-3q^2+4pq-p=0$ and a variable line px + qy =1 always touches a parabola whose axis is parallel to X-axis, then equation of the parabola is

1. 
2. 
3. 
4. 

Q. A line through the origin meets the circle $x^2+y^2=a^2$ at P and the hyperbola $x^2–y^2=a^2$ at Q. The locus of the point of intersection of the tangent at P to the circle and with the tangent at Q to the hyperbola is

1. None
2. 
3. 
4. 

Q.

What are the planes used in medicine and Anatomy?

Q.

1. Discontinuous at every points

2. Discontinuous exactly at three points

3. Discontinuous at only one point

4. Discontinuous exactly at two points

Q. The centre of the conic represented by the equation $x^2-6xy+y^2+6x+14y-2=0$ is

1. $(3,2)$
2. $(3,4)$
3. $(-3,4)$
4. $(5,4)$

Q. In the given figure $△OBC$ is an isosceles right angled triangle in which $AC$ is median, then which of the following is/are correct?

1. The minimum value of the quadratic expression corresponding to quadratic equation is $-\frac{1}{4}$
2. The number of solution of $f\left(|x|\right)=0$ is $4$.
3. $|b+c|=1$
4. The length of latus rectum of $f\left(x\right)$ is $1$.

Q. Why does SiO2 does not have a linear structure

Q. What does direction ratio actually mean???? Because the definition saying that 3 numbers proportional to the direction cosines of a line is called a direction ratio is not So imaginable ...

Q.

Match the columns by referring to the definition given below.

"A conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line”.

$\begin{array}{cc}\text{Definition}& \text{Name}\\ \text{P) Fixed point}& \text{1. Axis}\\ \text{Q) Fixed straight line}& \text{2. Vertex}\\ \text{R) Constant ratio}& \text{3. Directrix}\\ \text{S) Line passing through fixed point and perpendicular to fixed line}& \text{4. Focus}\\ & \text{5. Eccentricity}\end{array}$

1. P - 3, Q - 4, R - 5, S - 1

2. P - 4, Q - 3, R - 5, S - 2

3. P - 4, Q - 3, R - 5, S - 1

4. P - 4, Q - 3, R - 1, S - 5

Q. If $2p^2-3q^2+4pq-p=0$ and a variable line px+qy=1 always touches a parabola whose axis is parallel to X-axis, then equation of the parabola is

1. $(y-2{)}^2=24(x-4)$
2. $(y-4{)}^2=12(x-2)$
3. $(y-3{)}^2=12(x-1)$
4. $(y-4{)}^2=24(x-2)$

Q. The centre of the conic section $2x^2+4y^2+2xy+4x-6y+17=0$ is

1. $(\frac{11}{7},-\frac{8}{7})$
2. $(\frac{11}{7},\frac{8}{7})$
3. $(-\frac{11}{7},\frac{8}{7})$
4. $(-\frac{11}{8},\frac{7}{8})$

Q. What is RC and its dimension?

Q. If the tangent to the curve, $y=x^3+ax-b$ at the point $(1,-5)$ is perpendicular to the line, $-x+y+4=0$, then which one of the following points lies on the curve?

1. $(-2,1)$
2. $(2,-2)$
3. $(-2,2)$
4. $(2,-1)$

Q. The sum of the slopes of the tangents to the parabola $y^2=8x$ drawn from the point (–2, 3) is

1. 
2. 
3. 
4. 

Q.

Match the columns by referring to the definition given below.

"A conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line”.

$\begin{array}{cc}\text{Definition}& \text{Name}\\ \text{P) Fixed point}& \text{1. Axis}\\ \text{Q) Fixed straight line}& \text{2. Vertex}\\ \text{R) Constant ratio}& \text{3. Directrix}\\ \text{S) Line passing through fixed point and perpendicular to fixed line}& \text{4. Focus}\\ & \text{5. Eccentricity}\end{array}$

1. P - 3, Q - 4, R - 5, S - 1

2. P - 4, Q - 3, R - 5, S - 2

3. P - 4, Q - 3, R - 5, S - 1

4. P - 4, Q - 3, R - 1, S - 5

Q.

Which of the following describes a conic?

Here S is a fixed point and P is the moving point. e is the eccentricity of the conic

1. 

2. 

3. 

4. 

Q. Given below are Matching type questions, with two columns(each having some items) each.Each item of column$I$ has to be matched with the items of column$II$, by enclosing the correct match.
Note$:$An item of column$I$ can be matched with more than one items of column$II$.All the items of column $II$ have to be matched.
The equation of conics represented by the general equation of second degree $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$
and the discriminant of above equation is represented by $△$, where $△=abc+2fgh-a{f}^2-b{g}^2-c{h}^2$ or $\left(\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right)$
Now, match the entries from the following two columns

Q. Define Lateral Section?

Q.

A conic is the locus of a point such that its ratio of perpendicular distance from a fixed line to its distance from a fixed point is a constant.

1. False

2. True

Q. A straight line and a point not lying on it are given on a plane. Find the set of points which are equidistant from the given straight line and the given point.

Q. If the line $lx+my+n=0$ touches the parabola $y^2=4ax$, prove that $ln=a{m}^2$.

Q. The tangents at two points, P and Q, of a conic meet in T, and S is the focus ; prove that if the conic be a parabola, then $S{T}^2=SP.SQ.$

Q. The angle between the curves $x^3-3x{y}^2=2$ and $3x^{2}y-y^3=2$ is?

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

Q. From a point P perpendiculars PM and PN are drawn to curve $3x^2+4xy+y^2=0$ to meet at M and N. If MN$=2\sqrt{5}$, then?

1. Locus of P is pair of lines
2. Locus of P is circle
3. Area bounded by locus of P and x-axis above the x-axis is $50\pi$
4. Distance of any tangent to locus of P from origin is $10$ unit

Q. If $xy(y-x)=16,$ then $y$ has a stationary point when $x$ is equal to

Q. The angle between the lines $\frac{x-1}{1}=\frac{y-1}{1}=\frac{z-1}{2}\mathrm{and},\frac{x-1}{-\sqrt{3}-1}=\frac{y-1}{\sqrt{3}-1}=\frac{z-1}{4}$ is
(a) ${\cos }^{-1}\left(\frac{1}{65}\right)$

(b) $\frac{\mathrm{\pi }}{6}$

(c) $\frac{\mathrm{\pi }}{3}$

(d) $\frac{\mathrm{\pi }}{4}$

Q.

Match the columns by referring to the definition given below.

"A conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line”.

$\begin{array}{cc}\text{Definition}& \text{Name}\\ \text{P) Fixed point}& \text{1. Axis}\\ \text{Q) Fixed straight line}& \text{2. Vertex}\\ \text{R) Constant ratio}& \text{3. Directrix}\\ \text{S) Line passing through fixed point and perpendicular to fixed line}& \text{4. Focus}\\ & \text{5. Eccentricity}\end{array}$

1. P - 3, Q - 4, R - 5, S - 1

2. P - 4, Q - 3, R - 5, S - 2

3. P - 4, Q - 3, R - 5, S - 1

4. P - 4, Q - 3, R - 1, S - 5

Q. Find the angles at which the following vectors are inclined to each of the coordinate axes:
(i) $\hat{i}-\hat{j}+\hat{k}$

(ii) $\hat{j}-\hat{k}$

(iii) $4\hat{i}+8\hat{j}+\hat{k}$

Q. Assertion :In a right angle $\Delta ,\frac{a^2+b^2+c^2}{R}$ is always equal to $8$. Reason: $b^2=a^2+c^2$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion false but Reason is true