The equation x – y = 4 and x2 + 4xy + y2 = 0 represent the sides of
an equilateral triangle
Acute angle between the lines x2 + 4xy + y2 = 0 is
Angle bisectors of x2 + 4xy + y2 = 0 are given by
As x + y = 0 is perpendicular to x – y = 4, the given triangle is isosceles with vertical angle equal to π/3 and hence it is equilateral.
The line x + y = a, meets the axis of x and y at A and B respectively. A triangle AMN is inscribed in the triangle OAB, O being the origin, with right angle at N. M and N lie respectively on OB and AB. If the area of the triangle AMN is 3/8 of the area of the triangle OAB, then AN/BN is equal to.
The point (4, 1) undergoes the following transformation successively.
(i) Reflection about the line y = x
(ii) Translation through a distance 2 units along the positive direction of x-axis.
(iii) Rotation through an anlge π/4 about the origin in the anticlockwise direction
(iv) Reflection about x = 0
The final position of the given point is
A line cuts the x-axis at A (7, 0) and the y-axis at B(0, –5). A variable linePQ is draw perpendicular to AB cutting the x-axis at P and the y-axis at in θ. If AQ and BP intersect at R, the locus of R is
Equation of a line which is parallel to the line common to the pair of lines given by 6x2 – xy – 12y2 = 0 and 15x2 + 14xy – 8y2 = 0 and the sum of whose intercepts on the axes is 7, is
If pairs of lines x2 + 2xy + ay2 = 0 and ax2 + 2xy + y2 = 0 have exactly one line in common then the joint equation of the other two lines is given by
If the lines joining the origin to the intersection of the line y = mx + 2 and the curve x2 + y2 = 1 are at right angles, then
Let PQR be a right angled isosceles triangle right angled at P (2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is
If θ is an angle between the lines given by the equation 6x2 + 5xy – 4y2 + 7x + 13y – 3 = 0, then equation of the line passing through the point of intersection of these lines and making an angle θ with the positive x-axis is
If one of the lines given by the equation 2x2 + axy + 3y2 = 0 coincide with one of those given by 2x2 + bxy – 3y2 = 0 and the other lines represented by them be perpendicular, then
If the equation of the pair of straight lines passing through the point (1, 1), one making an angle θ with the positive direction of x-axis and the other making the same angle with the positive direction of y-axis is x2 – (a + 2)xy + y2 + a(x + y – 1) = 0,
a ≠ –2, then the value of sin 2θ is