General Equation of Conics

Q.

The equation ${(x+y)}^2-(x^2+y^2)=0$ represents

1. A circle

2. Two lines

3. Two parallel lines

4. Two mutually perpendicular lines

Q. The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point of intersection of the lines $7x+13y–87=0$ and $5x–8y+7=0$ and the latus rectum is $32\frac{\sqrt{2}}{5}$, then

1. $b^2=16$
2. $a^2=50$
3. $a^2=\frac{25}{2}$
4. $b^2=25$

Q. The equation $25(x^2+y^2-2x+1)=(4x-3y+1{)}^2$ represents

1. an ellipse
2. a parabola
3. a hyperbola
4. a pair of straight lines

Q. The equation $x^3-y{x}^2+x-y=0$ represents

1. a hyperbola and two straight lines
   
2. a straight line
   
3. a parabola and two straight lines
   
4. a straight line and a circle

Q. The conic represented by the equation $x^2+y^2+2xy+4x-6y+17=0$ is

1. a hyperbola
2. a parabola
3. a pair of straight lines
4. an ellipse

Q.

Match the column

$\begin{array}{cc}\text{Equation}& \text{Name of the curve}\\ \text{1)}{x}^2-2x-y-3=0& \text{P) Circle}\\ \text{2)}{x}^2+3xy+2y^2-x-4y-6=0& \text{Q) Parabola}\\ \text{3)}{x}^2+y^2-20=0& \text{R) Ellipse}\\ \text{4)}7x^2+7y^2+2xy+10x-10y+7=0& \text{S) Hyperbola}\\ \text{5)}6x^2-xy-y^2-23x+4y+15=0& \text{T) Pair of straight lines}\end{array}$

1. $1-T,2-Q,3-P,4-R,5-S$

2. $1-Q,2-T,3-P,4-R,5-S$

3. $1-Q,2-T,3-P,4-S,5-R$

4. $1-Q,2-S,3-P,4-T,5-R$

Q. A curve is such that the mid point of the segment of its normal intercepted between the point from where it is drawn and the point where the normal meets the $x-$axis lies on the parabola $2y^2=x.$ If the curve passes through origin, then the curve is

1. $y^2=2x+1-e^{2x}$
2. $y=x+1-e^{2x}$
3. $y^2=x+1-e^x$
4. $y=2x+1-e^x$

Q. The equation $25(x^2+y^2-2x+1)=(4x-3y+1{)}^2$ represents

1. an ellipse
2. a parabola
3. a hyperbola
4. a pair of straight lines

Q. The equation of the parabola whose focus lies at the intersection point of the lines $x+y=3$ and $x-y=1$ and directrix is $x-y+5=0$

1. $x^2+y^2+2xy+18x+6y-25=0$
2. $x^2+y^2+2xy+18x+6y+25=0$
3. $x^2+y^2+2xy-18x+6y-15=0$
4. $x^2+y^2+2xy-18x-6y+15=0$

Q.

Match the column

$\begin{array}{cc}\text{Equation}& \text{Name of the curve}\\ \text{1)}{x}^2-2x-y-3=0& \text{P) Circle}\\ \text{2)}{x}^2+3xy+2y^2-x-4y-6=0& \text{Q) Parabola}\\ \text{3)}{x}^2+y^2-20=0& \text{R) Ellipse}\\ \text{4)}7x^2+7y^2+2xy+10x-10y+7=0& \text{S) Hyperbola}\\ \text{5)}6x^2-xy-y^2-23x+4y+15=0& \text{T) Pair of straight lines}\end{array}$

1. $1-T,2-Q,3-P,4-R,5-S$

2. $1-Q,2-T,3-P,4-R,5-S$

3. $1-Q,2-T,3-P,4-S,5-R$

4. $1-Q,2-S,3-P,4-T,5-R$

Q. If the curve passing through $P(4,2),$ satisfying the differential equation $\frac{dy}{dx}=\frac{y}{x-y^2}$ is a conic, then which of the following statements is (are) correct?

1. Focus of the conic lies on line $y=2$.
2. Equation of directrix of conic is $4x–17=0$.
3. Eccentricity of the conic is $1$.
4. Eccentricity of the conic is $3$.

Q.

The equation of directrix of a conic is
$L:x+y-1=0$ and the focus is the point $(0,0)$. Find the equation of the conic if its eccentricity is
$\frac{1}{\sqrt{2}}$

1. $3x^2+3y^2+2xy+2x+2y-1=0$

2. $3x^2+3y^2+2xy+2x+2y+1=0$

3. $3x^2+3y^2+2xy-2x-2y+1=0$

4. $3x^2-3y^2+2xy+2x-2y+1=0$

Q. The equation $25(x^2+y^2-2x+1)=(4x-3y+1{)}^2$ represents

1. an ellipse
2. a parabola
3. a hyperbola
4. a pair of straight lines

Q. The locus of the point of intersection of the lines $\sqrt{3}kx+ky-4\sqrt{3}=0$ and $\sqrt{3}x-y-4\sqrt{3}k=0$ is a conic, whose length of latus rectum is equal to

Q. Which of the following is /are true for the curve $y=f\left(x\right)=2x^4-x^2$?

1. Curve is Symmetric about $x$ -axis
2. Curve touches the $x$ axis at origin
3. $y=f\left(x\right)$ has two points of local minima
4. Graph of $y=f\left(x\right)$ when $x\in [-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}]$ is

Q. If the curve passing through $P(4,2),$ satisfying the differential equation $\frac{dy}{dx}=\frac{y}{x-y^2}$ is a conic, then which of the following statements is (are) correct?

1. Focus of the conic lies on line $y=2$.
2. Equation of directrix of conic is $4x–17=0$.
3. Eccentricity of the conic is $1$.
4. Eccentricity of the conic is $3$.

Q. The solution of primitive integral equation $(x^2+y^2)dy=xy.dx$ is $y=y\left(x\right)$. If $y\left(1\right)=1$ and $y\left(x_0\right)=e$ then $x_0$ is

1. $\sqrt{2(e^2+1)}$
2. $\sqrt{3}e$
3. $\sqrt{2(e^2-1)}$
4. $\sqrt{\frac{1}{2}(e^2+1)}$

Q. Line $y=x$ cuts the curve $y^2=4ax$ at $A$ and $B$. Find equation of circle with $AB$ as diameter.

Q.

P is a moving point, S is the focus and L is the directrix as shown in figure.

Which of the following represents the equation of a conic?

1. $\frac{PS}{PM}=e$

2. $\frac{PM}{PS}=e$

3. $\frac{PM}{MS}=e$

4. $\frac{MS}{PM}=e$

Q. A straight line is such that the algebraic sum of the perpendiculars drawn upon it from any number of fixed points is zero. Then the straight line ______.

1. perpendicular to a fixed line
2. passes through a fixed point
3. is parallel to a fixed line
4. None of these

Q.

P is a moving point, S is the focus and L is the directrix as shown in figure.

Which of the following represents the equation of a conic?

1. 

2. 

3. 

4. 

Q.

The equation of directrix of a conic is
$L:x+y-1=0$ and the focus is the point $(0,0)$. Find the equation of the conic if its eccentricity is
$\frac{1}{\sqrt{2}}$

1. $3x^2-3y^2+2xy+2x-2y+1=0$

2. $3x^2+3y^2+2xy+2x+2y+1=0$

3. $3x^2+3y^2+2xy-2x-2y+1=0$

4. $3x^2+3y^2+2xy+2x+2y-1=0$

Q. There is a rectangle drawn such that their diagonals meet at the origin and their sides are parallel to the coordinate axes. The length of the sides parallel to y-axis is equal to the length of the conjugate axis of the hyperbola and the length of other two sides is equal to the distance between focii of the hyperbola. Then the area bounded by the hyperbola and the rectangle, is

1. $\frac{45}{8}-6\ln 2$
2. $\frac{45}{2}-24\ln 2$
3. $\frac{45}{8}+6\ln 2$
4. $\frac{45}{2}+24\ln 2$

Q. For the following planes, find the direction cosines of the normal to the plane and the distance of the plane from the origin.
$3y+5=0$.