The question asks to determine, identify, or convert between forms of a line's equation (slope-intercept, parametric, general form) or to find a line satisfying given conditions.
The number of integral values of $m$ so that the abscissa of point of intersection of lines $3 x + 4 y = 9$ and $y = m x + 1$ is also an integer, is: (1) 1 (2) 2 (3) 3 (4) 0
A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x - y - 2 = 0$ at the point $B$. If the length of the line segment $AB$ is $\frac { \sqrt { 29 } } { 3 }$, then $B$ also lies on the line (1) $2x + y = 9$ (2) $3x - 2y = 7$ (3) $x + 2y = 6$ (4) $2x - 3y = 3$
A line passing through the point $A ( 9,0 )$ makes an angle of $30 ^ { \circ }$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of $15 ^ { \circ }$ in the clockwise direction, then its equation in the new position is (1) $\frac { y } { \sqrt { 3 } - 2 } + x = 9$ (2) $\frac { x } { \sqrt { 3 } - 2 } + y = 9$ (3) $\frac { x } { \sqrt { 3 } + 2 } + y = 9$ (4) $\frac { y } { \sqrt { 3 } + 2 } + x = 9$
The equations of two sides AB and AC of a triangle ABC are $4 x + y = 14$ and $3 x - 2 y = 5$, respectively. The point $\left( 2 , - \frac { 4 } { 3 } \right)$ divides the third side BC internally in the ratio $2 : 1$. the equation of the side BC is (1) $x + 3 y + 2 = 0$ (2) $x - 6 y - 10 = 0$ (3) $x - 3 y - 6 = 0$ (4) $x + 6 y + 6 = 0$
Q66. The vertices of a triangle are $\mathrm { A } ( - 1,3 ) , \mathrm { B } ( - 2,2 )$ and $\mathrm { C } ( 3 , - 1 )$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is : (1) $x + y + ( 2 - \sqrt { 2 } ) = 0$ (2) $- x + y - ( 2 - \sqrt { 2 } ) = 0$ (3) $x + y - ( 2 - \sqrt { 2 } ) = 0$ (4) $x - y - ( 2 + \sqrt { 2 } ) = 0$
Q64. Let a variable line of slope $m > 0$ passing through the point $( 4 , - 9 )$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is (1) 30 (2) 25 (3) 15 (4) 10
3. (a) Write down the equation of the straight line through the point $( 1,2 )$ with slope - 1 . (b) Let $l$ be a line with equation $$y = ( 2 - a ) + a x$$ where $a$ is a constant. Show that, for any $a$, the line passes through the point $( 1,2 )$. Find the equation of the line perpendicular to this line which also passes through the point $( 1,2 )$. (c) Find the equations of the lines which pass through the point $( 1,2 )$ and have perpendicular distance 1 from the origin.
2. How many positive integers $n$ are there such that the line passing through points $A(-n, 0)$ and $B(0, 2)$ on the coordinate plane also passes through point $P(7, k)$, where $k$ is a positive integer? (1) 2 (2) 4 (3) 6 (4) 8 (5) Infinitely many
Consider the line $L: 5y + (2k-4)x - 10k = 0$ on the coordinate plane (where $k$ is a real number), and the rectangle $OABC$ with vertices at $O(0,0)$, $A(10,0)$, $B(10,6)$, $C(0,6)$. Let $L$ intersect the line $OC$ and the line $AB$ at points $D$ and $E$ respectively. Select the correct options. (1) When $k = 4$, the line $L$ passes through point $A$ (2) If the line $L$ passes through point $C$, then the slope of $L$ is $-\frac{5}{2}$ (3) If point $D$ is on the line segment $\overline{OC}$, then $0 \leq k \leq 3$ (4) If $k = \frac{1}{2}$, then the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary) (5) If the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary), then the slope of $L$ could be $\frac{3}{10}$
3. The perpendicular bisector of the line segment joining the points $( 2 , - 6 )$ and $( 5,4 )$ cuts the $x$-axis at the point with $x$-coordinate A $\frac { 1 } { 20 }$ B $\frac { 1 } { 6 }$ C $\frac { 1 } { 3 }$ D $\frac { 19 } { 5 }$ E $\frac { 41 } { 6 }$
In the rectangular coordinate plane, the line $y = \frac { x } { 7 }$ intersects the lines $x = 2$ and $x = 9$ at points $P$ and $R$ respectively. Accordingly, what is the length $| \mathrm { PR } |$ in units? A) $5 \sqrt { 2 }$ B) $6 \sqrt { 2 }$ C) $4 \sqrt { 10 }$ D) 8 E) 9
The square shown in the figure in the Cartesian coordinate plane is divided into two regions of equal area by a line with slope $\frac { - 1 } { 4 }$. If this line intersects the x-axis at point $(a, 0)$, what is a? A) 12 B) 14 C) 16 D) 18 E) 20
Emre marks a point on the x-axis of the Cartesian coordinate plane in a mathematics class activity. Then, by decreasing the x-coordinate of this marked point by 1 unit and increasing the y-coordinate by 3 units, he obtains a second point, and when he applies the same operation to the second point, he obtains a third point on the y-axis. What is the sum of the coordinates of the fourth point that Emre will obtain by applying the same operation to the third point? A) 4 B) 5 C) 6 D) 7 E) 8
In the rectangular coordinate plane, it is known that a line $d$ passes through point $A(-4, 1)$ and is perpendicular to the line $2x - y = 5$. If the point where line $d$ intersects the x-axis is $(a, 0)$ and the point where it intersects the y-axis is $(0, b)$, what is the sum $a + b$? A) -3 B) -1 C) 0 D) 1 E) 3
In the rectangular coordinate plane, points A and B lie on the line $y = x + 2$, and the distance between them is 3 units. Given that the coordinates of the midpoint of segment [AB] are $( -1, 1 )$, in which regions of the analytic plane are points A and B located? A) Both in region II B) Both in region III C) One in region I, the other in region II D) One in region I, the other in region III E) One in region II, the other in region III
In the rectangular coordinate plane, when point $A$ is translated 15 units in the negative direction along the $x$-axis, the resulting point lies on the line $d: 4x - 3y + 24 = 0$. Accordingly, if point $A$ is translated how many units in the positive direction along the $y$-axis, the resulting point will lie on line $d$? A) 9 B) 12 C) 16 D) 20 E) 25