Straight Lines-JEE-Main-PYQ’s

Mathematics Questions

Let $ABC$ be an equilateral triangle with orthocenter at the origin and the side $BC$ on the line $x + 2\sqrt{2}y = 4$ . If the co-ordinates of the vertex $A$ are $(\alpha, \beta)$ , then the greatest integer less than or equal to $|\alpha + \sqrt{2}\beta|$ is

(A) 5

(B) 4

(D) 3

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Let the angles made with the positive $x$ -axis by two straight lines drawn from the point $P(2, 3)$ and meeting the line $x + y = 6$ at a distance $\sqrt{\frac{2}{3}}$ from the point $P$ be $\theta_1$ and $\theta_2$ . Then the value of $(\theta_1 + \theta_2)$ is:

(A) $\frac{\pi}{2}$

(B) $\frac{\pi}{3}$

(D) $\frac{\pi}{6}$

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Let $A(1, 0)$ , $B(2, -1)$ and $C\left(\frac{7}{3}, \frac{4}{3}\right)$ be three points. If the equation of the bisector of the angle $ABC$ is $\alpha x + \beta y = 5$ , then the value of $\alpha^2 + \beta^2$ is

(A) 5

(B) 10

(D) 13

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Let $A(1, 2)$ and $C(-3, -6)$ be two diagonally opposite vertices of a rhombus, whose sides $AD$ and $BC$ are parallel to the line $7x - y = 14$ . If $B(\alpha, \beta)$ and $D(\gamma, \delta)$ are the other two vertices, then $|\alpha + \beta + \gamma + \delta|$ is equal to:

(A) 3

(B) 6

(D) 9

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A rectangle is formed by the lines $x = 0, y = 0, x = 3$ and $y = 4$ . Let the line $L$ be perpendicular to $3x + y + 6 = 0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $\left(\frac{1}{2}, -5\right)$ from the line $L$ is equal to:

(A) $\sqrt{10}$

(B) $2\sqrt{5}$

(D) $3\sqrt{10}$

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Among the statements
$(S1) :$ If $A(5, -1)$ and $B(-2, 3)$ are two vertices of a triangle, whose orthocentre is $(0, 0)$ , then its third vertex is $(-4, -7)$
and
$(S2) :$ If positive numbers $2a, b, c$ are three consecutive terms of an A.P., then the lines $ax + by + c = 0$ are concurrent at $(2, -2)$ ,

(A) both are incorrect

(B) only (S2) is correct

(D) only (S1) is correct

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Let a point $A$ lie between the parallel lines $L_1$ and $L_2$ such that its distances from $L_1$ and $L_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle $ABC$ , where the points $B$ and $C$ lie on the lines $L_1$ and $L_2$ , respectively, is :

(A) $21\sqrt{3}$

(B) $12\sqrt{2}$

(D) 27

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Let the area of the triangle formed by a straight line $L : x + by + c = 0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line $L$ makes an angle of $45^\circ$ with the positive $x$ -axis, then the value of $b^2 + c^2$ is:

(A) 90

(B) 83

(D) 97

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Let the triangle $PQR$ be the image of the triangle with vertices $(1, 3)$ , $(3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$ . If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$ , then $15(\alpha - \beta)$ is equal to:

(A) 21

(B) 19

(D) 24

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A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$ , respectively. If the locus of the point $P$ , that divides the rod $AB$ internally in the ratio $2 : 1$ is $9(x^2 + \alpha y^2 + \beta xy + \gamma x + 28y) - 76 = 0$ , then $\alpha - \beta - \gamma$ is equal to:

(A) 24

(B) 22

(D) 23

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Straight Lines-JEE-Main-PYQ’s

Mathematics Questions

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