Straight Lines and Pair of Straight Lines

The ends $A$, $B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ completed, then the locus of the foot of perpendicular drawn from $P$ to $A B$ is

The line parallel to the $x$-axis passing through the intersection of the lines $a x+2 b y+3 b=0$ and $b x-2 a y-3 a=0$ where $(a, b) \neq(0,0)$ is

Consider three points $P(\cos \alpha, \sin \beta), Q(\sin \alpha, \cos \beta)$ and $R(0,0)$, where $0<\alpha, \beta<\frac{\pi}{4}$. Then

If \(\mathrm{ABC}\) is an isosceles triangle and the coordinates of the base points are \(B(1,3)\) and \(C(-2,7)\). The coordinates of \(A\) can be

A square with each side equal to '\(a\)' above the \(x\)-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle \(\alpha\) \(\left(0<\alpha< \frac{\pi}{4}\right)\) with the positive direction of th...

Let \(\Gamma\) be the curve \(\mathrm{y}=\mathrm{be}^{-x / a}\) & \(\mathrm{L}\) be the straight line \(\frac{x}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1\) where \(\mathrm{a}, \mathrm{b} \in \mathbb{R}\).
Then

In \(\triangle \mathrm{ABC}\), co-ordinates of \(\mathrm{A}\) are \((1,2)\) and the equation of the medians through \(\mathrm{B}\) and C are \(x+\mathrm{y}=5\) and \(x=4\) respectively. Then the midpoint of \(\mathrm{BC}\) is

If \((1,5)\) be the midpoint of the segment of a line between the line \(5 x-y-4=0\) and \(3 x+4 y-4=0\), then the equation of the line will be

A rectangle ABCD has its side parallel to the line y = 2x and vertices A, B, D are on lines y = 1, x = 1 and x = \(-\)1 respectively. The coordinate of C can be

The locus of points (x, y) in the plane satisfying \({\sin ^2}x + {\sin ^2}y = 1\) consists of

The straight lines \(x + 2y - 9 = 0,3x + 5y - 5 = 0\) and \(ax + by - 1 = 0\) are concurrent if the straight line \(35x - 22y + 1 = 0\) passes through the point

If \(4{a^2} + 9{b^2} - {c^2} + 12ab = 0\), then the family of straight lines \(ax + by + c = 0\) is concurrent at

The equation \({r^2}{\cos ^2}\left( {\theta - {\pi \over 3}} \right) = 2\) represents

A, B are fixed points with coordinates (0, a) and (0, b) (a > 0, b > 0). P is variable point (x, 0) referred to rectangular axis. If the angle \(\angle\)APB is maximum, then

Consider the equation \(y - {y_1} = m(x - {x_1})\). If m and x1 are fixed and different lines are drawn for different values of y1, then

If the sum of the distances of a point from two perpendicular lines in a plane is 1 unit, then its locus is

If the algebraic sum of the distances from the points (2, 0), (0, 2) and (1, 1) to a variable straight line be zero, then the line passes through the fixed point

A ray of light along \(x + \sqrt 3 y = \sqrt 3\) gets reflected upon reaching X-axis, the equation of the reflected ray is

A moving line intersects the lines x + y = 0 and x \(-\) y = 0 at the points A, B respectively such that the area of the triangle with vertices (0, 0), A and B has a constant area C. The locus of the mid-point AB is given by the equation

Let A be the fixed point (0, 4) and B be a moving point on X-axis. Let M be the midpoint of AB and let the perpendicular bisector of AB meets the Y-axis at R. The locus of the midpoint P of MR is

If a > 0, b > 0 then the maximum area of the parallelogram whose three vertices are O(0, 0), A(a cos\(\theta\), b sin\(\theta\)) and B(a cos\(\theta\), \(-\) b sin\(\theta\)) is

The equation of the straight line passing through the point (4, 3) and making intercepts on the coordinate axes whose sum is \(- 1\) is

A line cuts the X-axis at A(7, 0) and the Y-axis at B(0, \(-\)5). A variable line PQ is drawn perpendicular to AB cutting the X-axis at P(a, 0) and the Y-axis at Q(0, b). If AQ and BP intersect at R, the locus of R is

Let each of the equations x2 + 2xy + ay2 = 0 and ax2 + 2xy + y2 = 0 represent two straight lines passing through the origin. If they have a common line, then the other two lines are given by

The equation \(r\,\cos \left( {\theta - {\pi \over 3}} \right) = 2\) represents

A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at P and Q respectively. The point O divides the segment PQ in the ratio

Straight lines x \(-\) y = 7 and x + 4y = 2 intersect at B. Points A and C are so chosen on these two lines such that AB = AC. The equation of line AC passing through (2, \(-\)7) is

A variable line passes through the fixed point \((\alpha ,\beta )\). The locus of the foot of the perpendicular from the origin on the line is

A straight line through the point (3, \(-\)2) is inclined at an angle 60\(^\circ\) to the line \(\sqrt 3 x + y = 1\). If it intersects the X-axis, then its equation will be

If the point of intersection of the lines 2ax + 4ay + c = 0 and 7bx + 3by \(-\) d = 0 lies in the 4th quadrant and is equidistant from the two axes, where a, b, c and d are non-zero numbers, then ad : bc equals to

A variable line passes through a fixed point \(({x_1},{y_1})\) and meets the axes at A and B. If the rectangle OAPB be completed, the locus of P is, (O being the origin of the system of axes).

A line cuts the X-axis at A(5, 0) and the Y-axis at B(0, \(-\)3). A variable line PQ is drawn perpendicular to AB cutting the X-axis at P and the Y-axis at Q. If AQ and BP meet at R, then the locus of R is

The point Q is the image of the point P(1, 5) about the line y = x and R is the image of the point Q about the line y = \(-\) X. The circumcentre of the \(\Delta\)PQR is

The area of the triangle formed by the intersection of a line parallel to X-axis and passing through P(h, k), with the lines y = x and x + y = 2 is h2. The locus of the point P is

The angular points of a triangle are A(\(-\) 1, \(-\) 7), B(5, 1) and C(1, 4). The equation of the bisector of the angle \(\angle\)ABC is

Transforming to parallel axes through a point (p, q), the equation \(2{x^2} + 3xy + 4{y^2} + x + 18y + 25 = 0\) becomes \(2{x^2} + 3xy + 4{y^2} = 1\). Then,

Let A(2, \(-\)3) and B(\(-\) 2, 1) be two angular points of \(\Delta\)ABC. If the centroid of the triangle moves on the line 2x + 3y = 1, then the locus of the angular point C is given by

Let d1 and d2 be the lengths of the perpendiculars drawn from any point of the line \(7x - 9y + 10 = 0\) upon the lines 3x + 4y = 5 and 12x + 5y = 7, respectively. Then,

The point P(3, 6) is first reflected on the line y = x and then the image point Q is again reflected on the line y = \(-\) x to get the image point Q'. Then, the circumcentre of the \(\Delta\)PQQ' is

The line through the points (a, b) and (\(-\)a, \(-\)b), passes through the point

The line AB cuts off equal intercepts 2a from the axes. From any point P on the line AB perpendiculars PR and PS are drawn on the axes. Locus of mid-point of RS is

The coordinates of a point on the line x + y + 1 = 0, which is at a distance \({1 \over 5}\) unit from the line 3x + 4y + 2 = 0, are

x + 8y \(-\) 22 = 0, 5x + 2y \(-\) 34 = 0, 2x \(-\) 3y + 13 = 0 are the three sides of a triangle. The area of the triangle is

Find the image of the point (\(-\)8, 12) with respect to the line 4x + 7y + 13 = 0.

The equation of the locus of the point of intersection of the straight lines \(x\sin \theta + (1 - \cos \theta )y = a\sin \theta\) and \(x\sin \theta - (1 + \cos \theta )y + a\sin \theta = 0\) is

If the three points A(1, 6), B(3, \(-\)4) and C(x, y) are collinear then the equation satisfying by x and y is

The coordinates of the two points lying on x + y = 4 and at a unit distance from the straight line 4x + 3y = 10 are

The equations of the lines through (1, 1) and making angles of 45\(^\circ\) with the line x + y = 0 are

The equations \(y = \pm \sqrt {3x}\), y = 1 are the sides of

If the three points (3q, 0) (0, 3p) and (1, 1) are collinear then which one is true?