Coordinate Geometry PYQ — Page 25

A parabola has the origin as its focus and the line $x=2$ as the directrix. Then the vertex of the parabola is at

The point diametrically opposite to the point $P(1,0)$ on the circle $x^2+y^2+2 x+4 y-3=0$ is

A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $1 / 2$. Then the length of the semi-major axis is

The perpendicular bisector of the line segment joining $P(1,4)$ and $Q(k, 3)$ has $y$-intercept - 4. Then a possible value of $k$ is

For the hyperbola $\frac{x^2}{\cos ^2 \alpha}-\frac{y^2}{\sin ^2 \alpha}=1$, which of the following remains constant when $\alpha$ varies?

The equation of a tangent to the parabola $y^2=8 x$ is $y=x+2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

Let $A(h, k), B(1,1)$ and $C(2,1)$ be the vertices of a right angled triangle with $A C$ as its hypotenuse. If the area of the triangle is $1$, then the set of values which ' $\mathrm{k}$ ' can take is given by

Let $C$ be the circle with centre $(0,0)$ and radius 3 units. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $\frac{2 \pi}{3}$ at its centre is

The locus of the vertices of the family of parabolas $y=\frac{a^3 x^2}{3}+\frac{a^2 x}{2}-2 a$ is

A straight line through the point A(3, 4) is such that its intercept between the axes is bisected at A. Its equation is

If the lines $3 x-4 y-7=0$ and $2 x-3 y-5=0$ are two diameters of a circle of area $49 \pi$ square units, the equation of the circle is

If $\left(a, a^2\right)$ falls inside the angle made by the lines $y=\frac{x}{2}, x>0$ and $y=3 x, x>0$, then a belongs to

The two lines $x=a y+b, z=c y+d ;$ and $x=a^{\prime} y+b^{\prime}, z=c^{\prime} y+d^{\prime}$ are perpendicular to each other if

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is

If the pair of lines $a x^2+2(a+b) x y+b y^2=0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then

An ellipse has $\mathrm{OB}$ as semi minor axis, $\mathrm{F}$ and $\mathrm{F}^{\prime}$ its focii and the angle FBF' is a right angle. Then the eccentricity of the ellipse is

Let $P$ be the point $(1,0)$ and $Q$ a point on the locus $y^2=8 x$. The locus of mid point of $P Q$ is

If a vertex of a triangle is $(1,1)$ and the mid-points of two sides through this vertex are $(-1,2)$ and $(3,2)$, then the centroid of the triangle is

If a circle passes through the point $(a, b)$ and cuts the circle $x^2+y^2=p^2$ orthogonally, then the equation of the locus of its centre is

If one of the lines given by $6 x^2-x y+4 c y^2=0$ is $3 x+4 y=0$, then $c$ equals

The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$. If one of the directrices is $x=$ 4 , then the equation of the ellipse is

A line makes the same angle $\theta$, with each of the $x$ and $z$ axis. If the angle $\beta$, which it makes with $y$-axis, is such that $\sin ^2 \beta=3 \sin ^2 \theta$, then $\cos ^2 \theta$ equals

The intercept on the line $y=x$ by the circle $x^2+y^2-2 x=0$ is $A B$. Equation of the circle on $\mathrm{AB}$ as a diameter is

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