Point Form of Normal: Ellipse

Q. Find the equations of the chords of the parabola y^2=4ax which pass through a point (-6a, 0) and which subtends an angle of 45` at the vertex.

Q. How many tangents to the circle $x^2+y^2=3$ are there which are normal to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$

1. 3
2. 2
3. 1
4. 0

Q. 14. ABC is a variable triangle such that A is (1, 2), and B and c lies on the line y=x+ (so is a variable).Then the locus of the orthocentre of triangle ABC is

Q.

What will be the new equation of the straight line 5x + 8y = 10, if the origin gets shifted to (2, -3)?

1. 5x – 8y = 14

2. 8x + 5y = 24

3. 5x + 8y = -4

4. 5x + 8y = - 14

Q.

What will be the new equation of the straight line 3x + 4y = 15, if the origin gets shifted to (1, -3) ?

1. 3x+4y=6

2. 4x+3y=6

3. 3x+4y=4

4. 3x-4y=4

Q.

The second degree equation $x^2+3xy+2y^2-x-4y-6=0$ represents a pair of lines.

1. True

2. False

Q. How many tangents to the circle $x^2+y^2=3$ are there which are normal to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$

1. 3
2. 2
3. 1
4. \N

Q. The equation $\frac{x^2}{2-\mathrm{\lambda }}+\frac{y^2}{\mathrm{\lambda }-5}+1=0$ represents an ellipse, if
(a) λ < 5
(b) λ < 2
(c) 2 < λ < 5
(d) λ < 2 or λ > 5

Q.

Find the equation of a normal to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=2$ at the point (4, 3).

1. 3x + 4y = 24

2. 4x - 3y = 7

3. 4x + 3y = 25

4. 3x - 4y = 0

Q. Solve the followings: (1) If the position vectors of A, B, C are respectively -5i+j , 5i+5j and 10i+7j . Then show that A, B, C are collinear. (2) If vector a= 2i-5j+3k and vector b= i-2j-4k , then find the value of |3a(vector)+2b(vector)|

Q. Reduce the following equations to the normal form and find p and α in each case:
(i) $x+\sqrt{3}y-4=0$
(ii) $x+y+\sqrt{2}=0$
(iii) $x-y+2\sqrt{2}=0$
(iv) x − 3 = 0
(v) y − 2 = 0.

Q. 89. Find the equation of the parabola whose focus is (2, 3) and whose directrix is 3x+4y=1

Q. Question) iii) Find the length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}$ = 1, whose middle point is $\left(\frac{1}{2},\frac{2}{5}\right)$.

Q. The eccentricity of an ellipse with center at the origin is $\frac{1}{2}$ if one of its directrices is x=-4, then the equation of the normal to it at $(1,\frac{3}{2})$ is

1. 2y-x=2
2. 4x-2y=1
3. 4x+2y=7
4. x+2y=4

Q. Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus-rectum of the hyperbola
(i) 9x² − 16y² = 144
(ii) 16x² − 9y² = −144
(iii) 4x² − 3y² = 36
(iv) 3x² − y² = 4
(v) 2x² − 3y² = 5.

Q. If R is a relation on the set A = [1, 2, 3, 4, 5, 6, 7, 8, 9] given by x R y ⇔ y = 3x, then R =

(a) [(3, 1), (6, 2), (8, 2), (9, 3)]
(b) [(3, 1), (6, 2), (9, 3)]
(c) [(3, 1), (2, 6), (3, 9)]
(d) none of these

Q. The normal at a point $P$ on the ellipse $x^2+4y^2=16$ intersects the $x$-axis at $Q$. If $M$ is the midpoint of the line segment $PQ$, then the locus of $M$ intersects the latus rectum of the given ellipse at points

1. $(\pm \frac{3\sqrt{5}}{2},\pm \frac{2}{7})$
2. $(\pm \frac{3\sqrt{5}}{2},\pm \frac{\sqrt{19}}{4})$
3. $(\pm 2\sqrt{3},\pm \frac{1}{7})$
4. $(\pm 2\sqrt{3},\pm \frac{4\sqrt{3}}{7})$

Q. If V and S are respectively the vertex and focus of the parabola y² + 6y + 2x + 5 = 0, then SV =
(a) 2
(b) 1/2
(c) 1
(d) none of these