Chord in terms of Midpoint

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If two distinct chords, drawn from the point $(p, q)$ on the circle $x^{2} + y^{2} = p x + q y$ (where $pq \neq = 0$ ) are bisected by the x-axis, then

p^{2} = q^{2}

(B)

p^{2} = 8 q^{2}

(C)

p^{2} < 8 q^{2}

(D)

p^{2} > 8 q^{2}

JEE Main 2021 (26 Aug Shift 2)

LEVELJEE Advanced

The locus of the mid points of the chords of the hyperbola $x^{2} - y^{2} = 4$ , which touch the parabola $y^{2} = 8 x$ , is :

(A)

y^{3} (x - 2) = x^{2}

(B)

x^{3} (x - 2) = y^{2}

(C)

y^{2} (x - 2) = x^{3}

(D)

x^{2} (x - 2) = y^{3}

JEE Main 2024 (04 Apr Shift 2)

LEVELJEE Main

Let $P Q$ be a chord of the parabola $y^{2} = 12 x$ and the midpoint of $P Q$ be at $(4, 1)$ . Then, which of the following point lies on the line passing through the points $P$ and $Q$ ?

(A)

(3, - 3)

(B)

(2, - 9)

(C)

(\frac{3}{2}, - 16)

(D)

(\frac{1}{2}, - 20)

JEE Main 2024 (29 Jan Shift 2)

LEVELJEE Main

Let $P (α, β)$ be a point on the parabola $y^{2} = 4 x$ . If $P$ also lies on the chord of the parabola $x^{2} = 8 y$ whose mid point is $(1, \frac{5}{4})$ . Then $(α - 28) (β - 8)$ is equal to ______.

JEE Main 2025 (January)

LEVELJEE Main

The length of the chord of the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ whose mid-point is $(1, \frac{1}{2})$ is:

(A)

\frac{5}{3} 15

(B)

\frac{1}{3} 15

(C)

\frac{2}{3} 15

(D)

15

JEE Main 2025 (January)

LEVELJEE Main

The equation of the chord, of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ whose mid-point is $(3, 1)$ is:

(A)

48 x + 25 y = 169

(B)

5 x + 16 y = 31

(C)

25 x + 101 y = 176

(D)

4 x + 122 y = 134

JEE Main 2025 (January)

LEVELJEE Main

If the midpoint of a chord of the ellipseis $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ is $(2, 4/3)$ and the length of the chord is $\frac{2 α}{3}$ then a is:

JEE Main 2025 (January)

LEVELJEE Main

If $α x + β y = 109$ is the equation of the chord of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1.$ whose mid point is $(\frac{5}{2}, \frac{1}{2})$ then $α + β$ is equal to:

If a chord, which is not a tangent, of the parabola $y^{2} = 16 x$ has the equation $2 x + y = p$ , and midpoint $(h, k)$ , then which of the following is(are) possible value(s) of $p, h$ and $k$ ?

(A)

p = 5, h = 4, k = - 3

(B)

p = - 1, h = 1, k = - 3

(C)

p = - 2, h = 2, k = - 4

(D)

p = 2, h = 3, k = - 4

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If two distinct chords, drawn from the point (p,q) on the circle x2+y2=px+qy (where pq=0) are bisected by the x-axis, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of the mid points of the chords of the hyperbola x2−y2=4, which touch the parabola y2=8x, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let PQ be a chord of the parabola y2=12x and the midpoint of PQ be at (4,1). Then, which of the following point lies on the line passing through the points P and Q?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P(α,β) be a point on the parabola y2=4x. If P also lies on the chord of the parabola x2=8y whose mid point is (1,45​). Then (α−28)(β−8) is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The length of the chord of the ellipse 4x2​+2y2​=1 whose mid-point is (1,21​) is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the chord, of the ellipse 25x2​+16y2​=1 whose mid-point is (3,1) is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the midpoint of a chord of the ellipseis 9x2​+4y2​=1 is (2​,4/3) and the length of the chord is 32α​​ then a is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If αx+βy=109 is the equation of the chord of the ellipse 9x2​+4y2​=1. whose mid point is (25​,21​) then α+β is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a chord, which is not a tangent, of the parabola y2=16x has the equation 2x+y=p, and midpoint (h,k), then which of the following is(are) possible value(s) of p,h and k?

If two distinct chords, drawn from the point $(p, q)$ on the circle $x^{2} + y^{2} = p x + q y$ (where $pq \neq = 0$ ) are bisected by the x-axis, then

The locus of the mid points of the chords of the hyperbola $x^{2} - y^{2} = 4$ , which touch the parabola $y^{2} = 8 x$ , is :

Let $P Q$ be a chord of the parabola $y^{2} = 12 x$ and the midpoint of $P Q$ be at $(4, 1)$ . Then, which of the following point lies on the line passing through the points $P$ and $Q$ ?

Let $P (α, β)$ be a point on the parabola $y^{2} = 4 x$ . If $P$ also lies on the chord of the parabola $x^{2} = 8 y$ whose mid point is $(1, \frac{5}{4})$ . Then $(α - 28) (β - 8)$ is equal to ______.

The length of the chord of the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ whose mid-point is $(1, \frac{1}{2})$ is:

The equation of the chord, of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ whose mid-point is $(3, 1)$ is:

If the midpoint of a chord of the ellipseis $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ is $(2, 4/3)$ and the length of the chord is $\frac{2 α}{3}$ then a is:

If $α x + β y = 109$ is the equation of the chord of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1.$ whose mid point is $(\frac{5}{2}, \frac{1}{2})$ then $α + β$ is equal to:

If a chord, which is not a tangent, of the parabola $y^{2} = 16 x$ has the equation $2 x + y = p$ , and midpoint $(h, k)$ , then which of the following is(are) possible value(s) of $p, h$ and $k$ ?