Straight Lines Questions (433)

The complete set of real values of 'a' such that the point \(P(a, \sin a)\) lies inside the triangle formed by the lines \(x - 2y + 2 = 0\), \(x + y = 0\) and \(x - y - p = 0\), is:

A pair of lines is given by \(ax^2 + 2hxy + by^2 = 0\). If point \((x_1, y_1)\) lies on one of the lines, find the relationship between the coefficients \(a\), \(b\), and \(h\).

Let B(1, –3) and D(0, 4) represent two vertices of rhombus ABCD in the (x, y) plane, then coordinates of vertex A if ∠BAD = 60° can be equal to:

If a straight line passing through the point P(- 3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is

The area of triangle PAB where P lies on the line x + y = 3, and A and B are the intercepts of another line with the axes, is given. Find the area of triangle PAB if |PA| = |PB| = 2.

Let $ABC$ be a triangle. Let $A$ be the point $(1, 2)$, $y = x$ is the perpendicular bisector of $AB$ and $x - 2y + 1 = 0$ is the angle bisector of angle $C$. If the equation of $BC$ is given by $ax + by - 5 = 0$ then the value of $a + b$ is ______.

The incentre of the triangle with vertices \((1,\sqrt{3})\), \((0,0)\) and \((2,0)\) is

A ray of light passing through the point \(A(1, 2)\) is reflected at a point \(B\) on the x-axis and then passes through \((5, 3)\). Then the equation of AB is:

Two equal sides $AB$ and $AC$ of an acute angle triangle $ABC$ are formed by the equations $7x - y + 3 = 0$ and $x + y - 3 = 0$, side $BC$ is $ax + by - 31 = 0$ where $a - 10b = 31$. Value of $2a + 10b + 31 = 0$ is ____.

In triangle $ABC$ if the median to side $BC$ has length $\left[11 - 6\sqrt{3}\right]^{\frac{1}{2}}$ and it divides angle $\angle A$ into angles $30°$ and $45°$. Then length of side $BC$ is __________.

In triangle ABC, the angle between lines AC and AB can be found using the formula for the angle between two lines. If the slope of AC is \(\frac{1}{4}\) and tan θ₁ = \(\frac{1}{3}\), find the slope m of AB such that tan θ₂ = \(\frac{1}{m}\).

Consider a trapezoid ABCD, one of whose non parallel sides AB which is 8 cm long is perpendicular to the base. The base BC and AD of trapezoid are 6 cm and 10 cm in lengths respectively. Let \(L_1, L_2, L_3, L_4\) represent the lines AB, BC, CD and DA respectively and \(d(P, L)\) denote the perpendicular distance of point P from line L.Find the area of region inside the trapezoid ABCD in which the point Q can lie satisfying \(d(Q, L_4) \leq d(Q, L_3)\):

The slopes of three sides of a triangle $ABC$ are $-1, -2, 3$ respectively. If the orthocenter of triangle $ABC$ is origin, then the locus of its centroid is $y = \frac{a}{b}x$ where $a, b$ are relatively prime than $b - a$ is equal to ________.

Three lines \(2x + 11y - 5 = 0\), \(24x + 7y - 20 = 0\) and \(4x - 3y - 2 = 0\) are given. Which of the following is correct?

Find the coordinates of point F given that \frac{BF}{CF} = \frac{BE}{AC} = \frac{a}{4} where B(0,a), C(a,a), and A(a,0).

If the slope of one of the line represented by $ax^2 + 2hxy + by^2 = 0$ is the square of the other, then the value of $\frac{a + b}{h} - \frac{8h^2}{ab}$ is ____.

Find the value of $c$ such that $2 \times \left(1 - \frac{c}{9}\right)(9 - c) = \frac{1}{2} \times 8 \times 1$.

The slopes of three sides of a triangle $ABC$ are $-1, -2, 3$ respectively. If the orthocenter of triangle $ABC$ is origin, then the locus of its centroid is $y = \frac{a}{b}x$ where $a, b$ are relatively prime then $b - a$ is equal to ____.

Let \(\triangle ABC\) be an isosceles triangle with \(AB = AC\). If \(AB : 4x + y = 7\), \(AC : x + 4y = 7\) and \(BC\) is passing through \((1, 1)\), then possible equation of \(BC\) is:

A line passes through the origin and point \((8, 11)\). Find the equation of this line.

The sides of a triangle are the straight lines $x + y = 1$, $7y = x$ and $3y + x = 0$. Then which of the following is an interior point of the triangle:

If one vertex of an equilateral triangle of side 'a' lies at origin and the other lies on the line $x - \sqrt{3}y = 0$, then the coordinates of the third vertex are:

A straight line through the point $A(-2, -3)$ cuts the lines $x + 3y = 9$ and $x + y + 1 = 0$ at $B$ and $C$ respectively. If $AB.AC = 20$, then product of slopes of line is____.

A possible equation of L is:

If the points where the lines $3x - 2y - 12 = 0$ and $x + ky + 3 = 0$ intersect both the coordinate axes are concyclic, then the number of possible real values of $k$ is:

In an acute triangle ABC, point H is the intersection point of altitude CE to AB and altitude BD to AC. A circle with DE as its diameter intersects AB and AC at points F and G respectively. If BC = 25, BD = 20 and BE = 7.The sum of the length of all the sides of △ABC is:

The circle 'S' touches the sides AB and AD of the rectangle ABCD and cuts the side DC at a single point F and the side BC at a single point E. If |AB| = 32, |AD| = 40 and |BE| = 1.The area of trapezoid AFCB is:

Let A(1, 1) and B(3, 3) be two fixed points and P be a variable point such that the area of ∆PAB remains constant equal to 1 for all positions of P. Then the locus of P is given by:

Perpendicular distance of the line AB from the point (2, 2) is:

A point on the straight line, \(3x + 5y = 15\) which is equidistant from the coordinate axes will lie only in

Two vertices of a triangle are \((0, 2)\) and \((4, 3)\) and its orthocenter is at the origin. Then the third vertex \(A(a, b)\) lies in which quadrant?

Find the y-intercept of the line passing through points with slope 2, where the point lies at distance 5 from (0, 2).

Let A = (0, 0), B = (4, 0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q.The locus of midpoints of all segments PQ as M varies along the segment AB is:

Given a triangle whose vertices are at \(0, 0\), \(4, 4\) and \(10, 0\). A square is drawn in it such that its base is on the x-axis and its two corners are on the 2 sides of the triangle. The area of the square is equal to:

What is the y-intercept of the line that is parallel to y = 3x, and which bisects the area of a rectangle with corners at (0, 0), (4, 0), (4, 2) and (0, 2)?

The base of an equilateral triangle is along the line given by \(3x + 4y = 9\). If a vertex of the triangle is \((1, 2)\), then the length of a side of the triangle is

Find the area of the pentagon \(ABCDE\) where \(A = (1, 3)\), \(B = (-2, 5)\), \(C = (-3, -1)\), \(D = (0, -2)\) and \(E = (2, 1)\).

A straight line through the origin $O$ meets the parallel lines $3x - 4y = 6$ and $6x - 8y + c = 0$ at points $Q$ and $P$ respectively such that $\left|\frac{OP}{OQ}\right| = \frac{4}{3}$. If $c = 2k$, then $k$ is equal to ______.

The point \((2a, a)\) lies on the line \(2x + 3y = 20\). The area of the triangle formed by the point \((a, a)\), origin, and point on the line is what?

Let $A(a,b)$, $B(3,4)$ and $(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3,7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is

In a rectangle with vertices at (0, 0), (a, 0), (a, b), and (0, b), if AD ⊥ BE where D and E are midpoints, find the relationship between a and b.

The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of an angle between the lines $L_1$ and $L_2$ is:

The lines $x + y = 0$, $-4y = 0$ and $2x - y = 0$ are the altitudes of a triangle. If one of the vertices has coordinates of the form $\lambda, -\lambda)$, if the locus of the centroid of such a triangle is $ax + by = 0$, then the value of $a + b$ is ______.

If the straight line through the point $P(3, 4)$ makes an angle $\frac{\pi}{6}$ with the $x$-axis and meets the line $12x + 5y + 10 = 0$ at $Q$ then the value of $\frac{(12\sqrt{3} + 5)}{11}$PQ is ______.

Two roads are represented by the equation \(y - x = 6\) and \(x + y = 8\). An inspection bungalow has to be so constructed that it is at a distance of 100 from each of the roads. Possible location of the bungalow is given by

Straight lines $2x + y = 5$ and $x - 2y = 3$ intersect at point A. Points B and C are chosen on these two lines such that $AB = AC$. Then the equation of a line BC passing through the point $(2, 3)$ is:

The x-coordinates of the vertices of a square of unit area are the roots of the equation $x^2 - 3|x| + 2 = 0$ and the y-coordinates of the vertices are the roots of equation $y^2 - 3y + 2 = 0$, then the possible vertices of the square is/are:

The orthocentre of a triangle is at origin and circumcentre is at (2, –3). Then the centroid of the triangle is:

Let A = (0,0), B = (4,0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q.The locus of midpoints of all segments PQ as M varies along the segment AB is:

$ABC$ is a triangle, whose vertex $A$ is $(3,4)$, $L_1 = 0$, $L_2 = 0$ are the angle bisectors of angle $B$ and $C$ respectively where $L_1 = x + 2y - 5 = 0$, $L_2 = x - 2y - 3 = 0$ also $AB = KAI$ where $I$ is the incentre then $K$ is ____.