Straight Lines

Let two points be $A(1,-1)$ $$ and $$ $B\left(\mathrm{0,2}\right)$ . If a point $P\left(x^{\mathrm{\text{'}}},y^{\mathrm{\text{'}}}\right)$ $\left({}^{}{}^{}\right)$ be such that the area of $?PAB=5$ $$ sq. units and it lies on the line, $3x+y-4\lambda =0$ $$ , then a value of $\lambda$ $$ is:

Let the equation of the pair of lines, y = px and y = qx, can be written as (y - px)(y - qx) = 0. Then the equation of the pair of the angle bisectors of the lines x² - 4xy – 5y² = 0 is:
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If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x² + y² = 1 is a circle of radius r, then r is equal to:

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of the line on the coordinate axes is $\frac{1}{4}.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> Three stones A, B and C are placed at the point (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is / are on the path of the man?

If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x - 2)² + (y – 3)² = 25 at the point (5, 7) is A, then 24A is equal to………………

Let a point P be such that its distance from the point (5,0) is thrice the distance of P from the point (-5,0). If the locus of the point P is a circle of radius r, then 4r² is equal to………………….

If the curve y = ax² + bx + c, $x\in R$ <!-- [if gte mso 9]><xml> </xml><![endif]--> passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are:

If the curves x = y⁴ and xy = k cut at right angles, then (4k)⁶ equal to………….

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be $\left(\frac{10}{3},\frac{7}{3}\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]-->. If a, b are the roots of the equation $a{x}^2+bx+1=0,$ <!-- [if gte mso 9]><xml> </xml><![endif]--> then the value of <!-- [if gte mso 9]><xml> </xml><![endif]-->  ${\alpha }^2+{\beta }^2-\alpha \beta$ is:

The locus of the point of intersection of the lines <!-- [if gte mso 9]><xml> </xml><![endif]-->= $\left(\sqrt[]{3}\right)kx+ky-4\sqrt[]{3}=0and\sqrt[]{3}x-y-4\left(\sqrt[]{3}\right)k$ 0 is a conic, whose eccentricity is…………………