Equation of a Chord When Its Mid Point Is Given

Q.

The locus of the midpoints of the focal chords of the parabola $y^2=4ax$ is

1. $y^2=2a(2x+a)$
2. $y^2=a(2x-a)$
3. $y^2=2a(x+a)$
4. $y^2=2a(x-a)$

Q. Two tangents on a parabola are $x-y=0$ and $x+y=0.$ Let $(2,3)$ is the focus of the parabola, then

1. The equation of the tangent at vertex is $4x-6y+5=0$
2. The equation of the tangent at vertex is $4x-6y+1=0$
3. The length of the latus rectum of the parabola is $\frac{10}{\sqrt{52}}$
4. The length of the latus rectum of the parabola is $\frac{10}{\sqrt{13}}$

Q.

Find the equation of chord of parabola
$y^2=4x$ whose midpoint is (4, 2)

1. $x-y+2=0$

2. $x+y+2=0$

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

4. $x-y-2=0$

Q. If the tangent at the point $P(2,4)$ to the parabola $y^2=8x$ meets the parabola $y^2=8x+5$ at $Q$ and $R$, then the mid-point of $QR$ is

1. $(2,4)$
2. $(4,2)$
3. $(7,9)$
4. none of these

Q. The circle passing through three distinct points $(1,t),(t,1)$ and $(t,t)$ for all values of $t$, also passes through the point:

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

Q.

From origin, chords are drawn to the circle $x^2+y^2-2y=0$.The locus of the middle points of these chords is

1. 

2. 

3. 

4. 

Q. The equation(s) of tangents drawn from the point $(1,4)$ to the parabola $y^2=12x$ is/are

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

Q.

The length and the midpoint of the chord 4x-3y+5=0 w.r.t circle $x^2+y^2-2x+4y-20=0$ is

1. 

2. 

3. 

4. 

Q.

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of the contact to the point (-4, -3). Find the equation of the curve given that it passes through (-2, 1).

Q. Tangents are drawn from the points on the parabola $y^2=-8(x+4)$ to the parabola $y^2=4x.$ Then the locus of mid-point of chord of contact of $y^2=4x$ is

1. $5y^2=-4(x+4)$
2. $5y^2=8(x+4)$
3. $5y^2=-4(x-4)$
4. $5y^2=8(x-4)$

Q. If the parabola $x^2=ay$ makes an intercept of length $\sqrt{40}$ unit on the line $y-2x=1$ then $a$ is equal to

1. $1$
2. $-2$
3. $-1$
4. $2$

Q. Let the line $4x+3y+1=0$ meets the parabola $8y^2=ax$ at $P,Q$. If the angle made by chord $PQ$ at the vertex of the parabola in $90°$, then $|a|=$

Q. PSQ is a focal choed of the ellipse $4x^2+9y^2=36$ such that SP= 4. If S is the another focus, write the value of SQ.

Q. The equation of chord to the parabola $y^2=4x$ whose sum of ordinates and product of abscissas of the endpoints of the chord is $4$ and $9$ respectively, is :

1. $2x-2y+6=0$
2. $2x-2y-6=0$
3. $3x-2y+3=0$
4. $3x-2y-3=0$

Q. Tangents are drawn from the points on the parabola $y^2=-8(x+4)$ to the parabola $y^2=4x.$ Then the locus of mid-point of chord of contact of $y^2=4x$ is

1. $5y^2=8(x+4)$
2. $5y^2=-4(x-4)$
3. $5y^2=8(x-4)$
4. $5y^2=-4(x+4)$

Q. PSQ is a focal chord of the parabola $y^2=8x$ .If $SP=6$ then $\frac{SP}{SQ}=$

1. $\frac{2}{3}$
2. $\frac{3}{4}$
3. $\frac{1}{9}$
4. $\frac{1}{4}$

Q. If two distinct chords of a parabola $y^2=ax$ passing through the point $(a,a)$ are bisected by the line $x+y=1,$ then the length of the latus rectum cannot be :

1. $3$
2. $4$
3. $5$
4. $6$

Q. Let the line $4x+3y+1=0$ meets the parabola $8y^2=ax$ at $P,Q$. If the angle made by chord $PQ$ at the vertex of the parabola in $90°$, then $|a|=$

Q. A tangent is drawn to the parabola $y^2=x$ at $(1,1)$, then the $x$ intercept of the tangent is

1. $1$
2. $-1$
3. $2$
4. $-\frac{1}{2}$

Q. if $(m_i,\frac{1}{m_i}),i=1,2,3,4$ are concyclic points then the value of $m_1{m}_2{m}_3{m}_4$ is

1. $1$
2. $-1$
3. $0$
4. $\infty$

Q.

Find the equation of chord of parabola
$y^2=4x$ whose midpoint is (4, 2)

1. $x-y+2=0$

2. $x+y+2=0$

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

4. $x-y-2=0$

Q. If the tangent at the point $P(2,4)$ to the parabola $y^2=8x$ meets the parabola $y^2=8x+5$ at $Q$ and $R$, then the mid-point of $QR$ is

1. $(4,2)$
2. $(2,4)$
3. $(7,9)$
4. none of these

Q. The point on the parabola $y^2=64x$ which is nearest to the line $4x+3y+35=0$ has coordinates

1. $(9,-24)$
2. $(-9,-24)$
3. $(1,81)$
4. $(4,-16)$

Q. The equation of a tangent to the parabola $y^2=9x$ from the point $(4,10)$ is

1. $x-4y+36=0$
2. $x-4y-36=0$
3. $81x-8y-162=0$
4. $9x-4y+4=0$

Q. Find the equations of tangents to parabola $y^2=4ax$ which are drawn from the point (2a, 3a).

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

Q. Find the equation of the tangent line to the curve $y=x^2-2x+7$ which is.
(a) parallel to the line $2x-y+9=0$.
(b) perpendicular to the line $5y-15x=13.$

Q. Find the equation of the tangent to the parabola $y=x^2-7x+3$, if that tangent is parallel to the straight line $5x+y-3=0.$

Q. The equation of the common tangent drawn to the curves $y^2=8x$ and $xy=-1$ is

1. $2y=x+6$
2. $y=x+2$
3. $3y=8x+2$
4. $y=2x+1$

Q. Find the locus of middle point of the chord of the parabola $y^2=4ax$ whose slope is 'm'

Q. The midpoint of the chord 2x – y – 2 = 0 of the parabola $y^2=8x$ is

1. (0, –2)
2. (3, 4)
3. 1, 0
4. (2, 2)