Chord with a Given Mid Point : Ellipse

Q. Chord of an ellipse are drawn through the positive end of the minor axis. Then their mid-point lies on

1. a circle
2. a parabola
3. an ellipse
4. a hyperbola

Q. If length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ whose middle point is $(\frac{1}{2},\frac{2}{5}),$ is $7\sqrt{\frac{k}{25}}$ unit, then $k=$

Q. Three distinct chords drawn from $(\alpha ,0)$ to the ellipse $x^2+2y^2=1.$ If these chords are bisected by the parabola $y^2=4x,$ then $\left[\alpha \right]=$
(where $[.]$ denotes greatest integer function)

Q. The locus of mid-points of focal chords of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $e$ is

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{ex}{a}$
2. None of the above
3. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{ex}{a}$
4. $x^2+y^2=a^2+b^2$

Q.

Find the equation of the chord of the ellipse $4x^2+25y^2=100$ whose middle point is (1, 1).

1. 4x + 25y = 29

2. 5x +25y = 30

3. 25x + 4y = 29

4. 25x + 5y = 30

Q.

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{5}{4}$ and $2x+3y-6=0$ is a focal chord of the hyperbola, then the length of the transverse axis is

1. $\frac{12}{5}$

2. $16$

3. $\frac{24}{7}$

4. $\frac{24}{5}$

5. $\frac{12}{7}$

Q. The equation of the line, passing through the centre and bisecting the chord $7x+y-1=0$ of the ellipse $\frac{x^2}{1}+\frac{y^2}{7}=1,$ is

1. $x-y=0$
2. $x+y=0$
3. $x-7y=0$
4. $7x-y=0$

Q. If the locus of the middle point of chords of an ellipse $\frac{x^2}{3}+\frac{y^2}{4}=1$ passing through $(2,0)$ is another ellipse $A,$ then the length of latus rectum of the ellipse $A$ is

1. $\frac{3}{8}$
2. $\sqrt{3}$
3. $\frac{1}{\sqrt{3}}$
4. $\frac{8}{3}$

Q.

On the ellipse $9x^2+25y^2=225,$ then find the point the distance from which to our focus $F_1$ is four times the distance to the other focus $F_2$

1. $\left(\frac{-15}{4},\frac{\sqrt{63}}{2}\right)$

2. $\left(\frac{-15}{4},\frac{\sqrt{63}}{4}\right)$

3. $\left(\frac{-15}{2},\frac{\sqrt{63}}{2}\right)$

4. $(-15,\sqrt{63})$

Q.

The locus of the midpoint of a chord of the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=4$ which subtends a right angle at the origin is

1. $\mathrm{x}+\mathrm{y}=2$

2. ${\mathrm{x}}^2+{\mathrm{y}}^2=1$

3. ${\mathrm{x}}^2+{\mathrm{y}}^2=2$

4. $\mathrm{x}+\mathrm{y}=1$

Q.

What is the equation of chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose middle point is $(x_1,y_1)$ ? You are given. $T=\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}-1and{S}_1\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}-1$

1. T + S₁=0

2. T = S₁

3. T + S₁= 1

4. T + 1= S₁

Q. Equation of the chord of the hyperbola $25x^2-16y^2=400$ which is bisected at the point (6, 2), is

1. 16x – 75y = 418
2. 75x - 16y = 418
3. 25x - 4y = 400
4. None of these

Q. If the locus of the middle point of chords of an ellipse $\frac{x^2}{3}+\frac{y^2}{4}=1$ passing through $(2,0)$ is another ellipse $A,$ then the length of latus rectum of the ellipse $A$ is

1. $\frac{8}{3}$
2. $\sqrt{3}$
3. $\frac{1}{\sqrt{3}}$
4. $\frac{3}{8}$

Q.

Find the equation of the chord of the ellipse $4x^2+25y^2=100$ whose middle point is (1, 1).

1. 4x +25y =29

2. 5x+25y=30

3. 25x +4y=29

4. 25x +5y =30

Q.

Find the equation of the chord of the ellipse $4x^2+25y^2=100$ whose middle point is (1, 1).

1. 4x +25y =29

2. 5x+25y=30

3. 25x +4y=29

4. 25x +5y =30

Q. The locus of mid-points of focal chords of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $e$ is

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{ex}{a}$
2. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{ex}{a}$
3. $x^2+y^2=a^2+b^2$
4. None of the above

Q. If the locus of the middle point of chords of an ellipse $\frac{x^2}{3}+\frac{y^2}{4}=1$ passing through $(2,0)$ is another ellipse $A,$ then the length of latus rectum of the ellipse $A$ is

1. $\frac{8}{3}$
2. $\sqrt{3}$
3. $\frac{1}{\sqrt{3}}$
4. $\frac{3}{8}$

Q.

What is the equation of chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose middle point is $(x_1,y_1)$ ? You are given. $T=\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}-1and{S}_1\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}-1$

1. $T+S_1=0$

2. $T=S_1$

3. $T+S_1=1$

4. $T+1=S_1$

Q. If length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ whose middle point is $(\frac{1}{2},\frac{2}{5}),$ is $7\sqrt{\frac{k}{25}}$ unit, then $k=$

Q. Locus of the mid point of the chord of circle $x^2+y^2=16$ which is subtending right angle at the point $(5,0)$ is

1. $x^2+y^2+4x+\frac{5}{2}=0$
2. $x^2+y^2-5x+\frac{9}{2}=0$
3. $x^2+y^2+5x+7=0$
4. $x^2+y^2+2x+5=0$

Q. Through a fixed point $(h,k)$ secants are drawn to the circle $x^2+y^2=r^2$ Then the locus of the midpoints of the chords intercepted by the circle is

1. $x^2+y^2=hx+ky$
2. $x^2-y^2=hx+ky$
3. $x^2+y^2=hx-ky$
4. $x^2-y^2=hx-ky$

Q. Chord of an ellipse are drawn through the positive end of the minor axis. Then their mid-point lies on

1. a circle
2. a parabola
3. an ellipse
4. a hyperbola

Q. The length of the chord of the parabola $y^2=x$ which is bisected at the point $(2,1)$ is

1. $5\sqrt{2}$
2. $4\sqrt{50}$
3. $2\sqrt{5}$
4. $4\sqrt{5}$

Q. Equation of the chord of the hyperbola $25x^2-16y^2=400$ which is bisected at the point (6, 2), is

1. 16x – 75y = 418
2. 75x - 16y = 418
3. 25x - 4y = 400
4. None of these

Q. Three distinct chords drawn from $(\alpha ,0)$ to the ellipse $x^2+2y^2=1.$ If these chords are bisected by the parabola $y^2=4x,$ then $\left[\alpha \right]=$
(where $[.]$ denotes greatest integer function)

Q. Equation of the chord of the hyperbola $25x^2-16y^2=400$ which is bisected at the point (6, 2), is

1. 16x – 75y = 418
2. 75x - 16y = 418
3. 25x - 4y = 400
4. None of these

Q. The equation of the chord of the parabola $y^2=8x$ which is bisected at the point $(2,-3)$, is.

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