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#### The length of the minor axis (along y-axis) of an ellipse in the standard form is  If this ellipse touches the line,  then its eccentricity is :  Option: 1   Option: 2   Option: 3   Option: 4

What is Ellipse? -

Ellipse

Standard Equation of Ellipse:

The standard form of the equation of an ellipse with center (0, 0) and major axis on the x-axis is

1. a > b

2.  the length of the major axis is 2a

3.  the coordinates of the vertices are (±a, 0)

4.  the length of the minor axis is 2b

5.  the coordinates of the co-vertices are (0, ±b)

-

Equation of Tangent of Ellipse in Parametric Form and Slope Form -

Slope Form:

-

Correct Option (2)

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#### If  are the eccentricities of the ellipse,  and the hyperbola,  respectively and is a point on the ellipse,  then  is equal to :  Option: 1 Option: 2 Option: 3 Option: 4

What is Ellipse? -

Ellipse

1. a > b

2.  the length of the major axis is 2a

3.  the coordinates of the vertices are (±a, 0)

4.  the length of the minor axis is 2b

5.  the coordinates of the co-vertices are (0, ±b)

-

What is Hyperbola? -

Hyperbola:

Eccentricity of Hyperbola:

-

So, k= 16

#### Let C the centroid of the triangle with vertices  Let P be the point of intersection of the lines  and  Then the line passing through the points C and P also passes through the point: Option: 1 Option: 2 Option: 3 Option: 4

Centroid -

Centroid

Centroid  of a triangle is the point of intersection of the medians of the triangle. A centroid divides the median in the ratio 2:1.

Whereas, the median is the line joining the mid-points of the sides and the opposite vertices.

The coordinates of the centroid of a triangle (G) whose vertices are A (x1, y1), B (x2, y2) and C(x3, y3), is given by

If D (a1, b1), E (a2, b2) and F (a3, b3) are the mid point of ΔABC, then its centroid is given by

-

Point of intersection of two lines -

Point of intersection of two lines

Equation of two non-parallel line is

If P (x1, y1) is a point of intersection of L1 and L2 , then solving these two equations of the line by cross multiplication

We get,

-

Equation of Straight Line (Part 2) -

Equation of Straight Line

(c) Two-point form

The equation of a straight line passing through the two given points (x1,y1) and  (x1,y1)is  given by

.

-

The centroid of triangle ABC D(2,2)

Point of intersection P

equation of line DP is  8x – 11y + 6 = 0

Point  (–9,–6) satisfies the equation

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#### If the image of the point P(1, −2, 3) in the plane, 2x+3y−4z+22=0 measured parallel to the line,     is Q, then PQ is equal to : Option: 1 Option: 5 $\sqrt{42}$ Option: 9 $6\sqrt{5}$ Option: 13 $3\sqrt{5}$

Intersection of line and plane -

Let the line

plane

intersect at P

to find P assume general point on line as

now put it in plane to find ,

-

Distance formula -

The distance between two points is =

- wherein

= -

PQ=2PM

Now, for finding M

Putting in equation of plane,

So,

#### The line of intersection of the planes and , is: Option: 1 $-2 \hat{i}+7 \hat{j}+13 \hat{k}$ Option: 2 $2 \hat{i}+7 \hat{j}-13 \hat{k}$ Option: 3 $-2 \hat{i}-7 \hat{j}+13 \hat{k}$ Option: 4 $-2 \hat{i}+7 \hat{j}+13 \hat{k}$

As we have learned

Equation of line as intersection of two planes -

Let the two intersecting planes be

and

then the parallel vector of line formed their intersection can be obtained by

and points can be obtained by putting and solving

and

say

Now the equation will be

-

putting z = 0

3x-y = 1    and   x + 4y = 2

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#### The tangent at the point (2, −2) to the curve, x2y2−2x=4(1−y) does not pass through the point :   Option: 1 (8,5) Option: 2 (4,1/3) Option: 3 (-2,-7) Option: 4 (-4,-9)

As learnt in concept ABCD

Differentiate both sides wrt.x.

at (2,-2)

Equation of tangent is

It doesn't pass through (-2,-7)

Concept ABCD

Slope of curve at a given point

To find slope, we differentiate with respect to x and find

Eg in

#### Let the normal at a point P on the curve $y^{2}-3x^{2}+y+10=0$ intersect the y-axis at $\left ( 0,\frac{3}{2} \right ).$ If m is the slope of the tangent at P to the curve, then $\left | m \right |$ is equal to Option: 1 4 Option: 2 3 Option: 3 2 Option: 4 1

Equation of Straight Line (Part 1) -

Equation of Straight Line

(b) Point-Slope form

Let the equation of give line l with slope ‘m’ is

y = mx + c    …..(i)

(x1,y1) lies on the line i

y1= mx1+c   ……(ii)

From (i) and (ii) [(ii) - (i)]

y - y= m( x - x1)

The equation of a straight line whose slope is given as ‘m’ and passes through the point (x1,y1) is  .

-

$\begin{array}{c}{2 y y^{\prime}+y^{\prime}-6 x=0} \\ {y^{\prime}=\frac{6 x}{2 y+1}} \\ {\frac{-1}{y^{\prime}}=\frac{-(2 y+1)}{6 x}}(\text{slope of the normal})\end{array}$

Equation of the normal $y-y_{1}=\frac{-\left(2 y_{1}+1\right)}{6 x_{1}}\left(x-x_{1}\right)$

Normal intersect at (0,3/2)

$\\\frac{3}{2}-y_{1}=\frac{-\left(2 y_{1}+1\right)}{6 x_{1}}\left(0-x_{1}\right)\\ 8y_1-8=0\\ y_1=1\\ x_1=\pm2\\ |m|=4$

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#### Let the line and the ellipse intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at and , then is equal to : Option: 1 Option: 2 Option: 3 Option: 4

Equation of Normal in Point Form and Parametric Form -

Equation of Normal in Point Form and Parametric Form

Point form:

-

Correct option (4)

#### The shortest distance between the lines and is :   Option: 1 Option: 2 Option: 3 Option: 4

Shortest Distance between Two Lines -

Distance between two skew lines

If L1 and L2  are two skew lines, then there is one and only one line perpendicular to each of lines L1 and L2 which is known as the line of shortest distance.

Vector form

If is the shortest distance vector between L1 and L2, then it being perpendicular to both and , therefore, the unit vector  along would be

where "d" is the magnitude of the shortest distance vector. Let θ be the angle between and .

Then

Hence, the required shortest distance is

-

=

Correct Option (4)