Length of Int...

Length of Intercept Made by a Circle on a Straight Line

Trending Questions

p x + q y = 40

is a chord of minimum length of the circle

(x - 10)^{2} + (y - 20)^{2} = 729.

If the chord passes through

(5, 15),

then

p^{2019} + q^{2019}

is equal to

$0$
$2$
$2^{2019}$
$2^{2020}$

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Q. If a chord of the circle

x^{2} + y^{2} = 8

makes equal intercepts

a

on the coordinate axes, then

$| a | < 2$
$| a | \leq 2 \sqrt{2}$
$| a | < 4$
$| a | < \sqrt{2}$

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Q. The sum of the squares of the lengths of the chords intercepted on the circle,

x^{2} + y^{2} = 16

by the lines,

x + y = n; n \in N

, where

N

is the set of all the natural numbers, is :

$105$
$210$
$320$
$160$

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The locus of the centre of the circle, which cuts the circle $x^{2} + y^{2} - 20 x + 4 = 0$ orthogonally and touches the line $x = 2$ , is

$x^{2} = 16 y$
$y^{2} = 4 x$
$y^{2} = 16 x$
$x^{2} = 4 y$

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Q. Two chords are drawn from the point

P (h, k)

on the circles

x^{2} + y^{2} - h x - k y = 0.

If the y-axis divides both the chords in the ratio

1 : 3,

then which of the following is/are correct?

$9 k^{2} > 16 h^{2}$
$k^{2} > 48 h^{2}$
$9 k^{2} \geq 16 h^{2}$
$4 k^{2} = h^{2}$

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The product of the length of the perpendiculars drawn from the point (1, 1) to the pair of lines x^2 + xy - 6y^2 = 0 is?

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Q. The equation of the circle which is touched by

y = x

, has its centre on the positive direction of the

x

-axis and cuts off a chord of length

2

units along the line

\sqrt{3} y - x = 0

, is

$x^{2} + y^{2} - 4 x + 2 = 0$
$x^{2} + y^{2} - 3 y + 2 = 0$
$x^{2} + y^{2} - 4 x - 3 y + 2 = 0$
$x^{2} + y^{2} = 2$

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Q. The shaded region shown in the figure is given by the inequation

$14 x + 5 y \geq 70 y \leq 14 and x - y \leq 5$
$14 x + 5 y \geq 70 y \leq 14 and x - y \geq 5$
$14 x + 5 y \leq 70 y \leq 14 and x - y \geq 5$
$14 x + 5 y \geq 70 y \geq 14 and x - y \geq 5$

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Choose the incorrect statement about the two circles whose equations are given below:

$x^{2} + y^{2} - 10 x - 10 y + 41 = 0$ and $x^{2} + y^{2} - 16 x - 10 y + 80 = 0$ .

Distance between two centers is the average radii of both the circles.
Circles have two intersection points.
Both circles’ centers lie inside the region of one another.
Both circles pass through the center of each other.

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Q. The equation of a line inclined at an angle

\frac{π}{4}

to the

x -

axis, such that the two circles

x^{2} + y^{2} = 4,

x^{2} + y^{2} - 10 x - 14 y + 65 = 0

intercept equal lengths on it, is

$2 x - 2 y + 3 = 0$
$2 x - 2 y - 3 = 0$
$x - y - 6 = 0$
$x - y + 6 = 0$

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If the lengths of the tangents from the point (1, 2) to the circles $x^{2} + y^{2} + x + y - 4 = 0$ and $3 x^{2} + 3 y^{2} - x - y - λ = 0$ are in the ratio 4:3 then $λ =$

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What is the equation of the chord of the circle $x^{2} + y^{2} = 81$ which is bisected at point $(- 2, 3)$ ?

$3 x - y = 13$
$3 x - 4 y = 13$
$2 x - 3 y = 13$
$3 x - 3 y = 13$
$2 x - 3 y = - 13$

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By joining the vertices of a 'n' sided polygon, pentagons are formed. Find the number of pentagons hence formed such that none of their side is common with the sides of the polygon.

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Q. The value of |c| for which the set

{(x, y) : x^{2} + y^{2} + 2 x \leq 1} \cup {(x, y) : x - y + c \geq 0}

contains only one point is equal to ___

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Q. Which of the following lines have the intercepts of equal lengths made by the circle

x^{2} + y^{2} - 2 x + 4 y = 0

is/are

$3 x - y = 0$
$x + 3 y = 0$
$x + 3 y + 10 = 0$
$3 x - y - 10 = 0$

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Let P and Q be two points denoting the complex numbers $α a n d β$ respectively on the complex plane. Which of the following equations can represent the equation of the circle passing through P and Q with least possible area ?

$a r g (\frac{z - α}{z - β}) = \frac{π}{2}$
$R e (z - α) (¯ ¯¯¯¯¯¯¯¯¯¯ ¯ z - β) = 0$
$| z - α |^{2} + | z - β |^{2} = (¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ α - β)^{2}$
$z ¯ z + (\frac{¯ α + ¯ β}{2}) z + (\frac{α + β}{2}) ¯ z + α ¯ β + ¯ α β = 0$

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Q. The equation of circle passing through the origin and cutting off equal intercepts of

2

units on the lines

\sqrt{3} y^{2} - \sqrt{3} x^{2} - 2 x y = 0

is/are

$(x + \sqrt{3}) (x - 1) + (y - \sqrt{3}) (y - 1) = 0$
$(x + 1) (x + \sqrt{3}) + (y + \sqrt{3}) (y - 1) = 0$
$(x - \sqrt{3}) (x + 1) + (y + 1) (y + \sqrt{3}) = 0$
$(x - \sqrt{3}) (x - 1) + (y + 1) (y - \sqrt{3}) = 0$

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The lines $a x^{2} + 2 h x y + b y^{2} = 0$ are equally inclined to the lines

$a x^{2} + 2 h x y + b y^{2} + λ (x^{2} + y^{2}) = 0$ for

= 1 only
= 2 only
for any value of
None of these

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Q. The number of integral values of

k

for which the chord of the circle

x^{2} + y^{2} = 125

passing through

P (8, k)

gets bisected at

P (8, k)

and has integral slope, is

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Q. The equation of a line parallel to the line

3 x + 4 y = 0

and touching the circle

x^{2} + y^{2} = 9

in the first quadrant is

$3 x + 4 y = 15$
$3 x + 4 y = 45$
$3 x + 4 y = 9$
$3 x + 4 y = 27$

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Let P, Q, R, S and T are five sets about the quadratic equation
(a – 5) $x^{2}$ – 2ax + (a – 4) = 0, a $\neq$ 5 such that
P : All values of ‘a’ for which the product of roots of given quadratic equation is positive.
Q : All values of ‘a’ for which the product of roots of given quadratic equation is negative.
R : All values of ‘a’ for which the product of real roots of given quadratic equation is positive.
S : All values of ‘a’ for which the roots of given quadratic are real.
T : All values of ‘a’ for which the given quadratic equation has complex roots.

none of the above.
least positive integer for set R is 2
greatest positive integer for set T is 3
least positive integer for set R is 3

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A line having slope 1 is drawn from a point A (-3, 0) cuts a curve y=xx²+x+1 at P and Q . Find l(AP) and l(AQ).

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Q. The equation of the circle which is touched by

y = x

, has its centre on the positive direction of the

x

-axis and cuts off a chord of length

2

units along the line

\sqrt{3} y - x = 0

, is

$x^{2} + y^{2} - 4 x + 2 = 0$
$x^{2} + y^{2} - 3 y + 2 = 0$
$x^{2} + y^{2} - 4 x - 3 y + 2 = 0$
$x^{2} + y^{2} = 2$

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Write the length of the intercept made oy the circle $x^{2} + Y^{2} + 2 x - 4 y - 5 = 0$
on y-axis.

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Using the method of integration find the area of the region bounded by lines:

2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0

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Let $L_{1}$ be a straight line passing through the origin and $L_{2}$ be the straight line x + y = 1. If the intercepts made by

the circle $x^{2} + y^{2} - x + 3 y = 0$ on $L_{1}$ and $L_{2}$ are equal, then which of the following equations can represent $L_{1}$

x + y = 0
x - y = 0
x + 7y = 0
x - 7y = 0

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Find the equation of chord centered at the point (1, 2) in the circle $x^{2} + y^{2} = 9.$

x + 2y - 5 = 0
x - 2y + 5 = 0
x + 2y + 5 = 0
x - 2y - 5 = 0

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The number of straight lines that are equally inclined to the three-dimensional co-ordinate axis, is

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Let $L_{1}$ be a straight line passing through the origin and $L_{2}$ be the straight line x + y = 1. If the intercepts made by

the circle $x^{2} + y^{2} - x + 3 y = 0$ on $L_{1}$ and $L_{2}$ are equal, then which of the following equations can represent $L_{1}$

x + y = 0
x - y = 0
x + 7y = 0
x - 7y = 0

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The length of the chord intercepted by the circle $x^{2} + y^{2} = r^{2}$ on the line $\frac{x}{a} + \frac{y}{b} = 1$

$\sqrt{\frac{r^{2} (a^{2} + b^{2}) - a^{2} b^{2}}{a^{2} + b^{2}}}$
$2 \sqrt{\frac{r^{2} (a^{2} + b^{2}) - a^{2} b^{2}}{a^{2} + b^{2}}}$
$2 \frac{\sqrt{r^{2} (a^{2} + b^{2}) - a^{2} b^{2}}}{a^{2} + b^{2}}$
$2 \frac{\sqrt{r^{2} (a^{2} + b^{2}) + a^{2} b^{2}}}{a^{2} + b^{2}}$

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