Straight Lines

Let the point $P(\alpha,\beta)$ be at a unit distance from each of the two lines $L_1:3x-4y+12=0$, and $L_2:8x+6y+11=0$. If P lies below $L_1$ and above $L_2$, then $100(\alpha+\beta)$ is equal to

A point P moves so that the sum of squares of its distances from the points (1, 2) and (−2, 1) is 14. Let $f(x,y)=0$ be the locus of P, which intersects the $x$-axis at the points A, B and the $y$-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to :

The equations of the sides AB, BC and CA of a triangle ABC are $2x+y=0$, $x+py=15a$ and $x-y=3$ respectively. If its orthocentre is $(2,a)$, $-\dfrac12<a<2$, then p is equal to ______.

The acute angle between the pair of tangents drawn to the ellipse $2x^2+3y^2=5$ from the point $(1,3)$ is

Let A(1, 1), B(−4, 3), C(−2, −5) be vertices of a triangle ABC, P be a point on side BC, and $\Delta_1$ and $\Delta_2$ be the areas of triangles APB and ABC, respectively. If $\Delta_1:\Delta_2=4:7$, then the area enclosed by the lines AP, AC and the $x$-axis is

The equations of the sides AB, BC and CA of a triangle ABC are $2x+y=0$, $x+py=39$ and $x-y=3$ respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true ?

If the length of the latus rectum of a parabola, whose focus is $(a,a)$ and the tangent at its vertex is $x+y=a$, is 16, then $|a|$ is equal to :

For $t\in(0,2\pi)$, if ABC is an equilateral triangle with vertices $A(\sin t,-\cos t)$, $B(\cos t,\sin t)$ and $C(a,b)$ such that its orthocentre lies on a circle with centre $\left(1,\dfrac13\right)$, then $\left(a^2-b^2\right)$ is equal to :

If the tangents drawn at the points P and Q on the parabola $y^2=2x-3$ intersect at the point R(0, 1), then the orthocentre of the triangle PQR is :

Let a line L pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $b\in\mathbf R-\left\{\frac43\right\}$. If the line L also passes through the point (1, 1) and touches the circle $17(x^2+y^2)=16$, then the eccentricity of the ellipse $\dfrac{x^2}5+\dfrac{y^2}{b^2}=1$ is :

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1, 1). If the line AP intersects the line BC at the point Q(k$_1$, k$_2$), then k$_1$ + k$_2$ is equal to :

Let the mirror image of a circle $c_1:\ x^2+y^2-2x-6y+\alpha=0$ in line $y=x+1$ be $c_2:\ 5x^2+5y^2+10gx+10fy+38=0$. If r is the radius of circle $c_2$, then $\alpha+6r^2$ is equal to _______.

Let $m_1$, $m_2$ be the slopes of two adjacent sides of a square of side a such that $a^2+11a+3(m_1^2+m_2^2)=220$. If one vertex of the square is $(10(\cos\alpha-\sin\alpha),\ 10(\sin\alpha+\cos\alpha))$, where $\alpha\in\left(0,\dfrac\pi2\right)$ and the equation of one diagonal is $(\cos\alpha-\sin\alpha)x+(\sin\alpha+\cos\alpha)y=10$, then $72(\sin^4\alpha+\cos^4\alpha)+a^2-3a+13$ is equal to :

If the system of linear equations

$\lambda x+y-z=-1$
$x-y-3z=2$
$-x+y+z=\mu$

has infinitely many solutions, then the equation of the line passing through the points $(\lambda+2\mu,\ 2\lambda+\mu)$ and $(1,\ \lambda\mu)$ is

Let A(0, a + 2), B(0, a), C(−2, 0) and D(2, 0) be four points and the lines AD and BC intersect at P(x, y). If the locus of P is the curve f(x, y) = 0 and the tangent at the point (4, γ) on this curve is $\dfrac x\alpha+\dfrac y\beta=1$, then α (β − γ) is equal to _______.

Consider a triangle $\Delta$ whose two sides lie on the $x$-axis and the line $x + y + 1 = 0$. If the orthocenter of $\Delta$ is $(1, 1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is

The area of the region
$$\left\{(x, y) : 0 \le x \le \frac94,\quad 0 \le y \le 1,\quad x \ge 3y,\quad x + y \ge 2\right\}$$
is

The value of $\lambda^2$ is ____.

The value of $D$ is ____.

Let $E$ be the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$. For any three distinct points $P$, $Q$ and $Q'$ on $E$, let $M(P, Q)$ be the mid-point of the line segment joining $P$ and $Q$, and $M(P, Q')$ be the mid-point of the line segment joining $P$ and $Q'$. Then the maximum possible value of the distance between $M(P, Q)$ and $M(P, Q')$, as $P$, $Q$ and $Q'$ vary on $E$, is ____.